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A committee of any size is to be formed from a group of 4 people. How many different committees can be formed?

Mathematics

ScanSolve AI Answer

Answer

15\text{15}

Step-by-step solution

4 steps

  1. Identify the total number of people and the possible committee sizes.

    There are 4 people in total.
    A committee can consist of one, two, three, or four members.

  2. Calculate the number of ways to form a committee for each possible size using combinations.

    The formula for combinations is C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n-k)!}, where nn is the total number of items, and kk is the number of items to choose.

    • For a committee of 1 member from 4 people:
    C(4,1)=4!1!(4−1)!=4!1!3!=4×3×2×1(1)(3×2×1)=4C(4, 1) = \frac{4!}{1!(4-1)!} = \frac{4!}{1!3!} = \frac{4 \times 3 \times 2 \times 1}{(1)(3 \times 2 \times 1)} = 4
    • For a committee of 2 members from 4 people:
    C(4,2)=4!2!(4−2)!=4!2!2!=4×3×2×1(2×1)(2×1)=244=6C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} = \frac{24}{4} = 6
    • For a committee of 3 members from 4 people:
    C(4,3)=4!3!(4−3)!=4!3!1!=4×3×2×1(3×2×1)(1)=4C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(1)} = 4
    • For a committee of 4 members from 4 people:
    C(4,4)=4!4!(4−4)!=4!4!0!=4!4!(1)=1C(4, 4) = \frac{4!}{4!(4-4)!} = \frac{4!}{4!0!} = \frac{4!}{4!(1)} = 1

  3. Sum the number of compositions for all possible committee sizes.

    Total number of different compositions = C(4,1)+C(4,2)+C(4,3)+C(4,4)C(4, 1) + C(4, 2) + C(4, 3) + C(4, 4)
    Total = 4+6+4+1=154 + 6 + 4 + 1 = 15

  4. Compare the result with the given options.

    The calculated total is 15. This matches option D.

    The final answer is 15\text{15}.

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