MAH-CET 2026 — Mathematics PYQ

Step 1: Determine whether to use Permutation or Combination

• In this problem, we need to form a committee of $3$ members from a pool of $7$ available people.

• Since the order in which the committee members are chosen does not matter (e.g., choosing Person A then Person B is the same as choosing Person B then Person A), this is a problem of selection rather than arrangement.

• Therefore, we use the Combination formula.

Step 2: Apply the Combination Formula

The standard formula for selecting $r$ objects out of $n$ distinct objects is given by ${}^n{C}_r$:

$${}^n{C}_r=\frac{n!}{r!(n-r)!}$$

From the given problem statement:

• Total number of people ($n$) = $7$

• Number of members to select ($r$) = $3$

Substitute these values into our formula:

$${}^7{C}_3=\frac{7!}{3!(7-3)!}$$

$${}^7{C}_3=\frac{7!}{3!\cdot 4!}$$

Step 3: Simplify the factorials

Expand the larger factorial in the numerator until it reaches the highest factorial present in the denominator:

$$7!=7\times 6\times 5\times 4!$$

Now substitute it back to cancel out $4!$:

$${}^7{C}_3=\frac{7\times 6\times 5\times 4!}{3!\times 4!}$$

$${}^7{C}_3=\frac{7\times 6\times 5}{3!}$$

Since $3!=3\times 2\times 1=6$:

$${}^7{C}_3=\frac{7\times 6\times 5}{6}$$

$${}^7{C}_3=7\times 5=35$$

Thus, there are $35$ distinct ways to form the committee.

Correct Answer

(b) $35$