Mock Test 1 — Permutations and Combinations

The total number of distinct 3-letter words that can be formed using the letters of the word `LOGARITHM` without repetition is:

Find the number of ways to arrange 5 people in a row such that 2 specific individuals, Amit and Bharati, never sit next to each other.

The value of the combinatorial expression $^{20}C_{17} + ^{20}C_{16}$ simplifies by Pascal's Identity to:

A committee of 4 members is to be selected from a group of 7 engineers. If the most senior engineer must always serve on the committee, the number of choices is:

The number of non-negative integer solutions to the linear equation $x + y + z = 5$ is exactly:

The number of ways to arrange the letters of the word `ALGEBRA` is:

If 6 distinct keys are arranged on a circular keyring, the total number of unique arrangements is:

The number of ways to choose a 3-member delegation from 6 people such that a specific person is never included is:

If $^nC_{12} = ^nC_{8}$, then the value of the parameter $n$ is:

The number of derangements of 3 items, where no item matches its original position index, is:

Find the number of positive integer solutions to the linear equation $x_1 + x_2 + x_3 = 6$ where $x_i \ge 1$.

Calculate the total number of rectangles contained inside a simple $2 \times 2$ grid square.

Find the total number of distinct rearrangements of the letters of the word `CAT` where every letter changes its initial slot position ($D_3$).

Assertion (A): The total number of subsets of a finite set with $n$ elements is given by $2^n$.
Reason (R): For each element in the set, there are exactly 2 independent choices: either it is included in the subset, or it is excluded.

Assertion (A): The expansion identity $^{n}C_r + ^{n}C_{r-1} = ^{n+1}C_r$ allows us to simplify compound summation strings.
Reason (R): Combining choices using Pascal's identity scales the population base by 1 while preserving the maximum target selection index $r$.