Permutations and Combinations - Class 11

Four distinct balls are to be distributed into three distinct boxes $B_1, B_2,$ and $B_3$. For each ball, a fair coin is tossed. If it shows heads, the ball is placed in $B_1$. If it shows tails, a fair die is rolled; if the die shows a number less than 5, the ball is placed in $B_2$, otherwise it is placed in $B_3$. What is the probability that no box remains empty?

Let $S$ be the set of all planes intersecting the coordinate axes at points $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$ such that $a, b, c \in \\{1, 2, 3, \\dots, 30\\}$. Let $L_k$ be the line passing through the point $(k, k, k)$ with direction ratios $(1, -2, 1)$. Find the number of planes in $S$ that contain the line $L_k$ for some integer $k \\ge 1$.

Let $S = \{1, 2, 3, 4, 5, 6\}$. The set $S$ is partitioned into 3 unordered subsets of 2 elements each. For each such partition, the sums of the elements in the three subsets are assigned to the upper triangular entries $a_{12}, a_{13}, a_{23}$ of a $3 \times 3$ skew-symmetric matrix $A$. Let $\mathcal{M}$ be the set of all distinct matrices $A$ that can be formed by considering all possible partitions of $S$ and all possible assignments of the sums. Find the value of $\sum_{A \in \mathcal{M}} \det(A \cdot \text{adj}(A) + 2I)$ where $I$ is the $3 \times 3$ identity matrix.

Let $N$ be the number of distinct arrangements of 3 identical red balls and 2 identical blue balls in a straight line such that the two blue balls are not adjacent. Let $f(x) = \lim_{t \to 0} \left( \frac{\sin(Ntx)}{Ntx} \right)^{\frac{1}{t^2}}$. If $g(x) = f(\tan^{-1}(\ln x))$, then the value of $g'(e)$ is

Let $S$ be a multiset containing 3 identical red balls, 2 identical blue balls, and 1 green ball. Let $a_r$ denote the number of ways to select exactly $r$ balls from $S$. Consider the differential equation: $x \left(\frac{d^m y}{dx^m}\right)^n + (m+1) \frac{d^{m-1} y}{dx^{m-1}} = a_5 x^{a_1 - 1}$ where $m = a_5 - 1$ and $n = a_6$. If $y(x)$ is a solution to this differential equation satisfying $y(1) = a_2$ and $y'(1) = a_3$, then the value of $40 y(2)$ is:

Let $a = \sum_{r=0}^{1} \binom{2}{r} \binom{2}{1-r}$ and $b = \sum_{r=0}^{2} (-1)^r \binom{2}{r} 2^{2-r}$. A multiset $S$ consists of exactly $a$ identical objects of type $A$ and $b$ identical objects of type $B$. Let $k$ be the total number of distinct linear arrangements of all the objects in $S$.

An experiment consists of $n$ independent trials, where $n$ is the constant term in the binomial expansion of $\left(x + \frac{1}{x}\right)^4$. In each trial, the probability of success is $p = \frac{b}{a+b}$.

If $P$ is the probability of obtaining at least $k$ successes in $n$ trials, then $P$ equals:

Let $m$ be a non-zero real number such that the quadratic equations $x^2 - 11x + m = 0$ and $x^2 - 14x + 2m = 0$ have a common root. Let $\alpha$ and $\beta$ be the non-common roots of the first and second equations, respectively. Let $S$ be the set of all integers $r$ satisfying the inequality $|r^2 - (\beta - 4)r - 4\alpha| \le 2\beta$ Let $N(r)$ denote the number of ways to distribute $r$ identical coins into $3$ distinct boxes (if such a distribution is not possible, $N(r)=0$). Find the value of $\sum_{r \in S} N(r)$.

Consider all permutations of the letters of the word $TARTAR$. For any permutation, let $x_k$ denote the alphabetical rank of the letter at the $k$-th position ($1 \le k \le 6$), where $A=1, R=2, T=3$. A permutation is defined as \textit{valid} if it satisfies the condition: $\sum_{k=1}^3 \sin^2\left(\frac{\pi}{2}(x_k - x_{k+3})\right) \neq 0$ Find the rank of the word $RATTAR$ among all valid permutations when they are arranged in dictionary order.

Let $S$ be the set of all distinct permutations of the multiset $\{1, 2, 3\}$. For each permutation $\sigma = (\sigma_1, \sigma_2, \sigma_3) \in S$, a plane $P_\sigma$ is defined by the equation $\frac{x}{\sigma_1} + \frac{y}{\sigma_2} + \frac{z}{\sigma_3} = 1$. Let $Q$ be the unique point of intersection of all the planes $P_\sigma$ for $\sigma \in S$. Let $N_1, N_2,$ and $N_3$ be the number of distinct permutations of the letters of the words 'TO', 'JEE', and 'NOON' respectively. A plane $\Pi$ contains the origin $O$, the point $Q$, and the point $R(N_1, N_2, N_3)$. A line $L$ lies in the plane $\Pi$, passes through the origin, and is perpendicular to the line $OQ$. If the direction ratios of $L$ are $(a, b, c)$ where $a, b, c$ are coprime integers and $a > 0$, find the value of $a^2 + b^2 + c^2$.

Let $S$ be a set of distinct beads. $A$ and $B$ are subsets of $S$ such that $n(A) = 8$, $n(B) = 6$, and $n(A \cap B) = 2$. All the beads that belong to exactly one of the sets $A$ or $B$ are selected. These selected beads are used to form two separate bracelets, each containing the same number of beads. Find the total number of distinct configurations of the pair of bracelets. (A configuration is invariant under the spatial rotation and flipping of individual bracelets, as well as the swapping of the two bracelets).

Let $A(x) = \begin{pmatrix} x-17 & 8 & 9 \\ 7 & x-16 & 9 \\ 7 & 8 & x-15 \end{pmatrix}$. Let $n$ be the non-zero repeated root of the equation $|A(x) \cdot A^T(x)| = 0$. If $n$ represents the number of ways to arrange $k$ distinct objects in a circle, find the value of $k^4 - 15$.

Let $k$ be the number of distinct ways to seat 5 people around a circular table such that two specific individuals are never adjacent. Consider the homogeneous function $H(x, y) = \sqrt{y^2 - \frac{k}{3} x^2}$. Let $L$ be the ray in the first quadrant consisting of the points where $H(x, y)$ is non-differentiable. A curve $y = f(x)$ (for $x \ge 1$) satisfies the differential equation $x \frac{dy}{dx} = y + x H\left(1, \frac{y}{x}\right)$ and intersects the ray $L$ at $x = 1$. Find the area of the region bounded by the curve $y = f(x)$, the ray $L$, and the vertical line $x = 3$.

Five married couples are to be seated around a circular table with $10$ identical chairs. In how many ways can they be seated such that men and women alternate, and no husband sits next to his own wife?

Let $k$ be the number of circular permutations of 7 distinct objects such that 4 specified objects are always together, and out of these 4, two specific objects are never adjacent to each other. A variable plane $\Pi$ makes non-zero intercepts $a, b, c$ on the $x, y, z$ axes respectively, such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{k}$. The plane $\Pi$ always passes through a fixed point $Q$. Let $L_1$ be the line given by the equations $x - 2y = 3$ and $y + 2z = 1$. A line $L_2$ passes through $Q$ and is parallel to $L_1$. Find the shortest distance between the line $L_2$ and the $z$-axis.

Six identical red balls and eight distinct blue balls are arranged randomly in a circle. Let $P$ be the probability that no two red balls are adjacent. Then the value of $P$ is:

Let $x_i$ be the number of ways to arrange $i+1$ identical items in $2i+3$ spaces in a row such that no two items are adjacent, for $i = 1, 2, 3$. Let $y_j$ be the number of ways to arrange $j+1$ identical items in $2j+2$ spaces in a row such that no two items are adjacent, for $j = 1, 2, 3$. If $M$ is a $3 \times 3$ matrix whose entries are given by $M_{i,j} = \binom{x_i + y_j}{2}$, find the value of $\det(M)$.

Let $m$ be the number of ways to arrange 5 identical black balls and 8 identical white balls in a row such that no two black balls are adjacent. Let $n$ be the number of such arrangements that are symmetric with respect to the center. Consider the function $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x) = \left(x^2 - \frac{m}{n}x + 114\right) |x - n|$ Let $S_c$ be the sum of all critical points of $f(x)$ and $S_s$ be the sum of all stationary points of $f(x)$. Find the value of $S_c + S_s$.

Let $S$ be the set of all integers $r$ such that the binomial expansion of $f_r(x) = \left( x^{\frac{1}{\sqrt{r}}} + x^{-\frac{1}{\sqrt{201-r}}} \right)^{201}$ contains a term independent of $x$. Find the number of elements in $S$.

Let $A = [a_{ij}]$ be a $4 \times 4$ matrix where $a_{ij}$ represents the number of ways to select $2$ distinct objects from a pool of $i+j$ distinct objects. Let $B = \text{adj}(\text{adj}(A))$. Find the value of $\det(B^3 + 2B^2 + 5I_4)$, where $I_4$ is the $4 \times 4$ identity matrix.

Let $A = \{-5, -4, -3, -2, -1, 1, 2, 3, 4, 5\}$. Find the number of ordered triplets $(a,b,c)$ such that $a \in A$, $b \in A$, $c \in \mathbb{Z}$, the line $ax + by + c = 0$ is tangent to the ellipse $9x^2 + 4y^2 = 36$, and the distance between this line and its parallel tangent is $2\sqrt{5}$.

In a continuous-time model of a combinatorial process, the sample space is partitioned into valid arrangements and total selections. The rate of generating valid arrangements is given by $a(t) = t(5 + 2\sin(t^2))$, while the rate of generating total selections is $s(t) = t(7 + 2\sin(t^2))$.

The cumulative measure of valid arrangements and total selections up to time $x$ are $A(x) = \int_0^x a(t) dt$ and $S(x) = \int_0^x s(t) dt$, respectively.

The asymptotic probability of an arrangement being valid is defined as $P = \lim_{x \to \infty} \frac{A(x)}{S(x)}$.

Find the value of $P$.

Let $S$ be the set of all $3 \times 3$ matrices $A$ of the form $A = \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix}$ where $a, b, c, d, e, f$ are non-negative integers such that $a+b+c+d+e+f = 10$. Find the number of matrices $A \in S$ such that the matrix $A^2 + \text{adj}(A)$ is singular.

Let $S$ be the set of all ordered triplets $(x, y, z)$ of non-negative integers satisfying the following conditions: $x + y + z \le 15$ $\frac{2}{x+y+z-4} \ge \frac{1}{x+y+z-9}$ Find the total number of elements in the set $S$.

A system distributes $n$ identical tasks to 3 distinct servers. The true probability that no server is idle is $P$, assuming each task is assigned independently and uniformly at random to one of the servers. A flawed model assumes that all possible valid configurations (non-negative integer distributions of the $n$ tasks among the 3 servers) are equally likely, yielding a probability $Q$ that no server is idle. If $\frac{P}{Q} = \frac{56}{27}$, what is the value of $n$?

A variable plane intersects the $x, y,$ and $z$ axes at exactly one point each, denoted by $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$ respectively, where $a, b,$ and $c$ are non-negative integers satisfying $a + b + c = 20$. Let $N$ be the number of such distinct planes. If $V$ is the maximum volume of the tetrahedron $OABC$ (where $O$ is the origin), find the value of $N + V$.

Let $S = \sum_{k=1}^{5} \frac{2k+1}{k^2(k+1)^2}$ and $n = \frac{1}{\sqrt{1 - S}}$. Find the number of integer solutions $(x_1, x_2, x_3, x_4)$ to the equation $x_1 + x_2 + x_3 + x_4 = n$ such that $|x_i| \le 5$ for all $i = 1, 2, 3, 4$.

Let $S_n$ be the set of all permutations of the set $\{1, 2, \dots, n\}$ where $n \ge 4$. A permutation $\pi$ is chosen uniformly at random from $S_n$. Let $X$ denote the number of fixed points of $\pi$ (i.e., the number of integers $i$ such that $\pi(i) = i$). What is the expected value of $X^3$, denoted by $E(X^3)$?

Let $n \ge 4$ be an integer. Let $A = [a_{ij}]$ be an $n \times n$ matrix where $a_{ij}$ is the number of derangements of the set $\{1, 2, \dots, n\}$ that map $i$ to $j$. Let $D_n$ be the total number of derangements of $n$ objects. If $I$ is the $n \times n$ identity matrix, which of the following represents the value of $\det(A) - \det(A - D_n I)$?

Let $D_k$ denote the number of derangements of $k$ distinct objects, with $D_0 = 1$. Define the sequence $S_n$ as: $S_n = \sum_{k=0}^{n} \binom{n}{k} D_k (-1)^{n-k}$ Which of the following represents the value of $\lim_{n \to \infty} \frac{S_n}{n!}$?

Let $S = \{1, 2, 3, 4, 5\}$. Find the number of permutations $\pi$ of $S$ such that $\pi(i) \neq i$ for all $1 \le i \le 5$, and additionally $\pi(1) \neq 2$, $\pi(2) \neq 3$, and $\pi(3) \neq 1$.

Let $P(x) = \begin{vmatrix} x-4 & -1 & -1 & -1 \\ -1 & x-4 & -1 & -1 \\ -1 & -1 & x-4 & -1 \\ -1 & -1 & -1 & x-4 \end{vmatrix}$. Let $n_1, n_2, n_3, n_4$ be the roots of the equation $P(x) = 0$. A set of $N = \sum_{i=1}^4 n_i$ distinct items is to be divided into four unlabeled groups such that the sizes of the groups are $n_1, n_2, n_3,$ and $n_4$. Let $W$ be the number of ways to perform this division. Find the value of $W$.

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$. A partition of $S$ into four disjoint subsets $A, B, C, D$, each of size 2, is chosen. Let $s_A, s_B, s_C, s_D$ denote the sum of the elements in $A, B, C, D$ respectively. A function $f: [0, 4] \to \mathbb{R}$ is defined as: $f(x) = \begin{cases} 2x + s_A & 0 \le x < 1 \\ x^2 + s_B & 1 \le x < 2 \\ 3x + s_C & 2 \le x < 3 \\ 4x + s_D & 3 \le x \le 4 \end{cases}$ Find the number of ordered quadruplets of subsets $(A, B, C, D)$ such that there exists a continuous injective function $g: \mathbb{R} \to \mathbb{R}$ for which the composite function $h(x) = g(f(x))$ is continuous on $[0, 4]$.

Let $N$ be the number of ways to divide a set of 8 distinct points into two equal groups. Consider two skew lines $L_1$ and $L_2$ in $\mathbb{R}^3$, where $L_1$ passes through the point $(N, 0, 0)$ and is parallel to the vector $3\hat{i} - 2\hat{j}$, and $L_2$ passes through the point $(0, N, 0)$ and is parallel to the vector $3\hat{i} - \hat{k}$. The shortest distance between the lines $L_1$ and $L_2$ is:

A researcher is distributing $6$ distinct energy modules and $14$ identical cooling units into $4$ distinct server racks. Each rack must receive at least $1$ energy module and at least $2$ cooling units to function properly. Due to spatial limitations, no server rack can contain more than $10$ items in total (energy modules and cooling units combined). Find the total number of valid distributions.

Let $A$ be a matrix of order $3 \times 3$ such that $|A| = 2$. Let $\lambda \in \mathbb{R}$ be the real root of the equation $|-\lambda A| + \lambda^2 |A| = 24$ Consider the following set $S$ of 5 linear equations in variables $x, y, z$: \begin{align*} E_1&: x + 2y + 3z = 1 \ E_2&: 2x + 3y + 4z = 2 \ E_3&: 3x + 4y + 5z = 3 \ E_4&: 4x + 5y + 6z = \lambda^2 \ E_5&: 5x + 6y + 7z = \lambda + 7 \end{align*} Find the number of non-empty subsets of $S$ that form a consistent system of equations.

A dataset $S$ is formed by the following process: For every possible non-empty selection of items from a collection of 3 identical red, 3 identical blue, and 3 identical green balls, the total number of balls in the selection is recorded as a data point in $S$. Let $\delta$ be the minimum possible value of the mean deviation of $S$ about any real number $\alpha$. Find the value of $63 \delta$.

Consider the set of functions $S = \{f_1, f_2, f_3, f_4, f_5\}$ from $\mathbb{R}$ to $\mathbb{R}$ defined as: $f_1(x) = -e^x, \quad f_2(x) = x^3 + x, \quad f_3(x) = x^4 + x^2$ $f_4(x) = -x^5 - x, \quad f_5(x) = \ln(e^x + 1)$ A non-empty subset of $S$ is chosen and its elements are composed in decreasing order of their indices to form a new function $H(x)$. (For example, if $\{f_1, f_3, f_4\}$ is chosen, $H(x) = f_4(f_3(f_1(x)))$). Find the number of such subsets for which $H(x)$ is strictly increasing on $\mathbb{R}$.

Let $S = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ be a set of vectors where $\vec{v}_k = (2k-1)\hat{i} + 4\hat{j} + (8-k)\hat{k}$. The total number of ways to select a non-empty subset of vectors from $S$ is $1023$. Let $\Pi_1$ be the plane containing the vectors $\vec{v}_1$ and $\vec{v}_2$, and $\Pi_2$ be the plane containing the vectors $\vec{v}_3$ and $\vec{v}_4$. If $\theta$ is the acute angle between the planes $\Pi_1$ and $\Pi_2$, evaluate the expression: $E = \frac{\cos \theta}{\binom{n}{2}} \sum_{1 \le i < j \le n} \frac{|\vec{v}_i|^2 |\vec{v}_j|^2 - (\vec{v}_i \cdot \vec{v}_j)^2}{(j-i)^2} + \sin \theta \sum_{k=1}^n |\vec{v}_k|^4$

Let $N$ be the total number of ways to select at least one fruit from a basket containing exactly 2 identical apples and 2 identical bananas. Let a function $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{vmatrix} \int_0^x (t-1)^N dt & \int_0^x (t-1)^{N+1} dt & \int_0^x (t-1)^{N+2} dt \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix}$ Let $m$ and $n$ be the number of local maxima and local minima of $f(x)$, respectively. Find the ordered pair $(m, n)$.

Let $N$ be the number of ways to select at least one fruit from a basket containing 2 identical apples, 3 identical mangoes, and 4 identical oranges. Let $P(x) = \frac{1}{N} \sum_{k=0}^{N} k^2 \binom{N}{k} x^k (1-x)^{N-k}, \quad x \in \mathbb{R}$ If the solution set of the inequality $\left[ P(x) - 26x \right] \le 1$ is $x \in (a, b)$, where $[\cdot]$ denotes the greatest integer function, then the value of $b - a$ is:

Four distinct energy states can each hold a maximum of $5$ identical quantum particles. $14$ such identical particles are distributed among these $4$ states. Let $S$ be the sample space of all valid distributions. A distribution is selected at random from $S$. Let $E$ be the event that the selected distribution has \textit{exactly one} energy state containing exactly $5$ particles.

If this selection process is repeated independently with replacement, what is the probability that the $3^{\text{rd}}$ occurrence of event $E$ happens exactly on the $6^{\text{th}}$ selection?

Let $a$ be the number of ways to select 4 items from 3 distinct types of items (assuming an unlimited supply of each type). Let $b$ be the number of ways to distribute 5 identical balls into 3 distinct boxes such that each box receives at least 1 ball. Let $c$ be the number of ways to select 2 items from 4 distinct types of items (assuming an unlimited supply of each type). Consider the system of linear equations: $\begin{aligned} a_{11}x + a_{12}y + a_{13}z &= d_1 \\ a_{21}x + a_{22}y + a_{23}z &= d_2 \\ a_{31}x + a_{32}y + a_{33}z &= d_3 \end{aligned}$ where the coefficient matrix $M = [a_{ij}]$ is a skew-symmetric matrix with $a_{12} = a$, $a_{13} = b$, and $a_{23} = c$. If the system has infinitely many solutions, the vector $\vec{d} = d_1 \hat{i} + d_2 \hat{j} + d_3 \hat{k}$ must be orthogonal to which of the following vectors?

Let $N$ be the number of ways to select 4 items from a collection of 3 identical apples, 4 identical oranges, and 5 identical bananas. Let $P$ be a plane with normal vector $\vec{n} = \vec{a} \times \vec{b}$, where $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$. The plane $P$ satisfies the equation $\vec{r} \cdot \vec{n} = N$. Let $L_1$ be the line $\frac{x-2}{1} = \frac{y+1}{2} = \frac{z-1}{-1}$, and let $L_2$ be the orthogonal projection of the line $L_1$ onto the plane $P$. If $L_3$ is the line $\frac{5x-2}{5} = \frac{5y-4}{5} = \frac{z+1}{1}$, find the shortest distance between the lines $L_2$ and $L_3$.

Let $N$ be the number of ways to select 4 items from a collection comprising 3 identical apples, 3 identical bananas, and 3 identical cherries. Let $I = \int_{1/e}^{e^{N/6}} \frac{|\ln x|}{x^2} dx$. Which of the following inequalities holds true?

Let $f(x) = \frac{10}{x^2+1}$ and $K = \{ [f(x)] : x \in [1, 2] \}$, where $[ \cdot ]$ denotes the greatest integer function. For each $k \in K$, a set $S_k$ consisting of exactly $k$ distinct elements is given. Let $W_k$ be the number of ordered pairs $(A, B)$ of subsets of $S_k$ such that $A \cap B \neq \emptyset$. Find the value of $\sum_{k \in K} W_k$.

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Two subsets $A$ and $B$ of $S$, each containing exactly 4 elements, are chosen independently and uniformly at random. Let $E$ be the event that $1 \in A \cap B$ or $2 \in A \cap B$. Given that $A \cap B \neq \emptyset$, the probability of event $E$ occurring is $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Find the value of $m+n$.

Two students independently select a subset of items from a common pool of $n$ distinct items. Let $k$ be the number of ways they can make their selections such that their chosen subsets have at least one item in common.

A point $P$ divides the line segment joining $A(3, -10, -2)$ and $B\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ internally in the ratio $k:1$. If the coordinates of $P$ represent the direction cosines of a line, find the value of $n$.

Let $S = \{1, 2, 3, \dots, 20\}$. Three sets $A, B,$ and $C$ are defined as follows: $A = \{x \in S \mid \text{sgn}(x^2 - 15x + 50) \le 0\}$ $B = \{x \in S \mid \text{sgn}(x^2 - 21x + 104) \le 0\}$ $C = \{x \in S \mid \text{sgn}(x^2 - 12x + 27) \le 0\}$ where $\text{sgn}(\cdot)$ denotes the signum function. Find the number of ways to select 3 distinct elements from $S$ such that the selection contains at least one element from $A \cap B$ and at least one element from $C$.

Let $A$ be a $3 \times 3$ matrix whose characteristic equation is $x^3 - c_1 x^2 + c_2 x - c_3 = 0$. The roots of this equation are $\lambda_1, \lambda_2,$ and $\lambda_3$, which are given by: $\lambda_1 = \sum_{n=1}^{60} E_2(n!), \quad \lambda_2 = \sum_{n=1}^{75} E_3(n!), \quad \lambda_3 = \sum_{n=1}^{100} E_5(n!)$ where $E_p(k)$ denotes the exponent of the highest power of prime $p$ dividing $k$. Find the value of $c_1$.

Let $V$ be the maximum absolute value among all terms in the binomial expansion of $(5 + 2x)^{100}$ for a given integer $x$. Let $S$ be the set of all integers $x$ such that $E_3(V) = 4$, where $E_p(N)$ denotes the exponent of the highest power of prime $p$ dividing $N$. Find the value of $\sum_{x \in S} x^2$.