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
Here are the solutions for question 1: 1. How many ways can you form a 4-person committee from 6 men and 5 women? a) with no restrictions We need to choose 4 people from a total of 6 + 5 = 11 people. Since the order does not matter, we use combinations. Step 1: Calculate the total number of people. Total people = 6 men + 5 women = 11 people Step 2: Use the combination formula nk = (n!)/(k!(n-k)!) to choose 4 people from 11. 114 = (11!)/(4!(11-4)!) = (11!)/(4!7!) = (11 × 10 × 9 × 8 × 7!)/(4 × 3 × 2 × 1 × 7!) = (11 × 10 × 9 × 8)/(4 × 3 × 2 × 1) = 11 × 10 × 3 = 330 The number of ways is 330. b) the committee has 2 men and 2 women We need to choose 2 men from 6 men AND 2 women from 5 women. Step 1: Choose 2 men from 6 men. 62 = (6!)/(2!(6-2)!) = (6!)/(2!4!) = (6 × 5 × 4!)/(2 × 1 × 4!) = (30)/(2) = 15 Step 2: Choose 2 women from 5 women. 52 = (5!)/(2!(5-2)!) = (5!)/(2!3!) = (5 × 4 × 3!)/(2 × 1 × 3!) = (20)/(2) = 10 Step 3: Multiply the number of ways to choose men by the number of ways to choose women. 15 × 10 = 150 The number of ways is 150. c) the committee has only 1 woman If the committee has 1 woman, and it's a 4-person committee, then the remaining 4 - 1 = 3 members must be men. Step 1: Choose 1 woman from 5 women. 51 = (5!)/(1!(5-1)!) = (5!)/(1!4!) = (5 × 4!)/(1 × 4!) = 5 Step 2: Choose 3 men from 6 men. 63 = (6!)/(3!(6-3)!) = (6!)/(3!3!) = (6 × 5 × 4 × 3!)/(3 × 2 × 1 × 3!) = (120)/(6) = 20 Step 3: Multiply the number of ways to choose women by the number of ways to choose men. 5 × 20 = 100 The number of ways is 100.