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: Determine the possible compositions of the committee that satisfy the condition "at least 2 women". A committee of 4 members must contain at least 2 women. This means the possible compositions are: • Exactly 2 women and 2 men • Exactly 3 women and 1 man • Exactly 4 women and 0 men Step 2: Calculate the number of ways for each case using combinations. The formula for combinations is nk = (n!)/(k!(n-k)!). • Case 1: 2 women and 2 men Number of ways to choose 2 women from 5: 52 = (5!)/(2!(5-2)!) = (5 × 4)/(2 × 1) = 10 Number of ways to choose 2 men from 7: 72 = (7!)/(2!(7-2)!) = (7 × 6)/(2 × 1) = 21 Total ways for Case 1 = 10 × 21 = 210 • Case 2: 3 women and 1 man Number of ways to choose 3 women from 5: 53 = (5!)/(3!(5-3)!) = (5 × 4 × 3)/(3 × 2 × 1) = 10 Number of ways to choose 1 man from 7: 71 = (7!)/(1!(7-1)!) = 7 Total ways for Case 2 = 10 × 7 = 70 • Case 3: 4 women and 0 men Number of ways to choose 4 women from 5: 54 = (5!)/(4!(5-4)!) = (5 × 4 × 3 × 2)/(4 × 3 × 2 × 1) = 5 Number of ways to choose 0 men from 7: 70 = (7!)/(0!(7-0)!) = 1 Total ways for Case 3 = 5 × 1 = 5 Step 3: Sum the ways from all possible cases to find the total number of ways to form the committee. Total number of ways = Ways for Case 1 + Ways for Case 2 + Ways for Case 3 Total ways = 210 + 70 + 5 = 285 The committee can be formed in 285 ways. Got more? Send 'em!