This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 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. Step 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)!), where n is the total number of items, and k 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)) = 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)) = (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)) = 4 • For a committee of 4 members from 4 people: C(4, 4) = (4!)/(4!(4-4)!) = (4!)/(4!0!) = (4!)/(4!(1)) = 1 Step 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) Total = 4 + 6 + 4 + 1 = 15 Step 4: Compare the result with the given options. The calculated total is 15. This matches option D. The final answer is 15. Last free one today — make it count tomorrow, or type /upgrade for unlimited.