Question

The number of ways a committee of 4 people can be chosen from a panel of 10 people is

• A. 315
• B. 240
• C. 210
• D. 720
• E. 120

Hint:

Combinatorics Tip: If a problem mentions "committee", "group", or "team" with no specific roles (like President/VP), it is ALWAYS a combination ($C$). If roles are assigned, it's a permutation ($P$).

Answer 1

The Correct Option is C

Concept:
When forming a committee or group where the order of selection does not matter, we use the combination formula. The number of ways to choose $r$ items from a total pool of $n$ items is: $${}^nC_r = \frac{n!}{r!(n-r)!}$$

Step 1: Identify the total pool size (n) and selection size (r).
From the problem description: Total available people ($n$) = $10$ People needed for the committee ($r$) = $4$

Step 2: Set up the combination formula.
We need to calculate "10 choose 4": $${}^{10}C_4 = \frac{10!}{4!(10-4)!}$$

Step 3: Simplify the factorials.
Subtract the terms in the denominator to simplify: $${}^{10}C_4 = \frac{10!}{4! \times 6!}$$

Step 4: Expand the factorials for cancellation.
Expand the larger factorial in the numerator ($10!$) until it reaches the larger factorial in the denominator ($6!$) so they cancel out: $${}^{10}C_4 = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4 \times 3 \times 2 \times 1 \times 6!}$$ Cancel the $6!$: $${}^{10}C_4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}$$

Step 5: Perform the final arithmetic division.
Notice that $4 \times 2 = 8$, which cancels the $8$ in the numerator: $${}^{10}C_4 = \frac{10 \times 9 \times 7}{3}$$ Divide $9$ by $3$ to get $3$: $${}^{10}C_4 = 10 \times 3 \times 7 = 210$$ Hence the correct answer is (C) 210.