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 Exercise 1: Exercise 1: How many ways can you form a 4-person committee from 6 men and 5 women? Total number of people = 6 men + 5 women = 11 people. The committee size is 4 people. a) With no restrictions Step 1: Determine the total number of people (n) and the committee size (k). n = 11 (total people) k = 4 (committee members) Step 2: Use the combination formula, as the order of selection does not matter for a committee. C(n, k) = nk = (n!)/(k!(n-k)!) Step 3: Substitute the values and calculate. C(11, 4) = (11!)/(4!(11-4)!) = (11!)/(4!7!) C(11, 4) = (11 × 10 × 9 × 8 × 7!)/(4 × 3 × 2 × 1 × 7!) C(11, 4) = (11 × 10 × 9 × 8)/(4 × 3 × 2 × 1) C(11, 4) = (7920)/(24) C(11, 4) = 330 The number of ways to form a 4-person committee with no restrictions is 330. b) the committee has 2 men and 2 women Step 1: Determine the number of ways to choose 2 men from 6 men. C(6, 2) = (6!)/(2!(6-2)!) = (6!)/(2!4!) C(6, 2) = (6 × 5 × 4!)/(2 × 1 × 4!) = (30)/(2) = 15 Step 2: Determine the number of ways to choose 2 women from 5 women. C(5, 2) = (5!)/(2!(5-2)!) = (5!)/(2!3!) C(5, 2) = (5 × 4 × 3!)/(2 × 1 × 3!) = (20)/(2) = 10 Step 3: Multiply the number of ways to choose men and women, as these are independent selections. Total ways = C(6, 2) × C(5, 2) = 15 × 10 = 150 The number of ways to form a committee with 2 men and 2 women is 150. c) the committee has only 1 woman Step 1: If the committee has only 1 woman, and the committee size is 4, then the remaining 3 members must be men. Determine the number of ways to choose 1 woman from 5 women. C(5, 1) = (5!)/(1!(5-1)!) = (5!)/(1!4!) C(5, 1) = (5 × 4!)/(1 × 4!) = 5 Step 2: Determine the number of ways to choose 3 men from 6 men. C(6, 3) = (6!)/(3!(6-3)!) = (6!)/(3!3!) C(6, 3) = (6 × 5 × 4 × 3!)/(3 × 2 × 1 × 3!) = (120)/(6) = 20 Step 3: Multiply the number of ways to choose women and men. Total ways = C(5, 1) × C(6, 3) = 5 × 20 = 100 The number of ways to form a committee with only 1 woman is 100.