From a group of 8 people, 5 have skin rashes. If 5 people are selected randomly, what is the probability that exactly 3 have skin rashes and none have skin rashes?

You're on a roll — Here's the solution for question 12 using the combination method. Given: Total number of people examined = 8 Number of people with skin rashes = 5 Number of people without skin rashes = 8 - 5 = 3 Number of people selected randomly = 5 The total number of ways to select 5 people from 8 is given by 85. 85 = (8!)/(5!(8-5)!) = (8!)/(5!3!) = (8 × 7 × 6)/(3 × 2 × 1) = 56 a) Exactly 3 had skin rashes. Step 1: Determine the number of ways to select exactly 3 people with skin rashes. This means selecting 3 people from the 5 with rashes AND 5-3=2 people from the 3 without rashes. Number of ways to select 3 with rashes: 53 = (5!)/(3!2!) = (5 × 4)/(2 × 1) = 10 Number of ways to select 2 without rashes: 32 = (3!)/(2!1!) = 3 The number of favorable ways is the product of these combinations: Favorable ways = 53 × 32 = 10 × 3 = 30 Step 2: Calculate the probability. P(exactly 3 rashes) = Favorable waysTotal ways = (30)/(56) = (15)/(28) Step 3: Convert the probability to three decimal places. (15)/(28) ≈ 0.53571... P(exactly 3 rashes) ≈ 0.536 b) None had skin rashes. Step 1: Determine the number of ways to select none with skin rashes. This means selecting 0 people from the 5 with rashes AND 5-0=5 people from the 3 without rashes. Number of ways to select 0 with rashes: 50 = 1 Number of ways to select 5 without rashes: 35 = 0 (It is impossible to choose 5 people from a group of 3). The number of favorable ways is the product of these combinations: Favorable ways = 50 × 35 = 1 × 0 = 0 Step 2: Calculate the probability. P(none with rashes) = Favorable waysTotal ways = (0)/(56) = 0 Step 3: Convert the probability to three decimal places. P(none with rashes) = 0.000 c) At most 2 had skin rashes. Step 1: Determine the number of ways to select at most 2 people with skin rashes. This means the number of people with rashes can be 0, 1, or 2. • Case 1: 0 rashes (and 5 without rashes) Favorable ways = 50 × 35 = 1 × 0 = 0 • Case 2: 1 rash (and 4 without rashes) Number of ways to select 1 with rashes: 51 = 5 Number of ways to select 4 without rashes: 34 = 0 Favorable ways = 51 × 34 = 5 × 0 = 0 • Case 3: 2 rashes (and 3 without rashes) Number of ways to select 2 with rashes: 52 = (5 × 4)/(2 × 1) = 10 Number of ways to select 3 without rashes: 33 = 1 Favorable ways = 52 × 33 = 10 × 1 = 10 The total number of favorable ways for "at most 2 rashes" is the sum of these cases: Total favorable ways = 0 + 0 + 10 = 10 Step 2: Calculate the probability. P(at most 2 rashes) = Total favorable waysTotal ways = (10)/(56) = (5)/(28) Step 3: Convert the probability to three decimal places. (5)/(28) ≈ 0.17857... P(at most 2 rashes) ≈ 0.179 Drop the next question.