Mock Test 2 — Permutations and Combinations

Find the number of ways to arrange 4 boys and 4 girls in a row such that the boys and girls sit alternating.

The number of non-negative integer solutions to the equation $x + y + z + w = 12$ such that $x \ge 1, y \ge 2, z \ge 0, w \ge 0$ is:

Find the rank of the short word `RANK` when all its structural permutations are arranged alphabetically as in a dictionary.

The number of ways to divide 8 distinct objects into 4 equal groups of 2 objects each is:

There are 5 distinct fountain pens and 5 distinct caps. The number of ways to pair them such that every single pen gets a wrong cap is:

The total number of rectangles contained inside a standard chessboard grid ($8 \times 8$ squares) is:

The sum of all 4-digit numbers that can be formed using the digits \{1, 2, 3, 4\} without repetition is:

The number of ways to choose 4 letters from the word `EXAMINATION` is a multi-case problem. The total number of selection options is:

If 6 people are to be seated around a round table, and 2 specific individuals refuse to sit next to each other, the number of valid arrangements is:

The value of the combinatorial expression $^{n}C_0 + ^{n}C_1 + ^{n}C_2 + \dots + ^{n}C_n$ represents the total sum of binomial coefficients, which equals:

Find the number of distinct ways to arrange the letters of the word `PERMUTATION` such that the relative order of the vowels remains completely unchanged.

Find the number of divisors of the integer 360, excluding 1 and the number itself.

Determine the number of solutions to the equation $x_1 + x_2 + x_3 = 7$ in positive integers ($x_i \ge 1$).

Assertion (A): The number of ways to arrange $n$ distinct items in a circle is $(n-1)!$.
Reason (R): Fixing the position of one item breaks the rotational symmetry of the circle, transforming the remaining choices into a standard linear permutation of $n-1$ items.

Assertion (A): The number of non-negative integer solutions to $x+y=10$ is exactly 11.
Reason (R): The variable $x$ can take any integer value from 0 up to 10 independently, and each choice uniquely determines the value of $y$ since $y = 10 - x$.