How many ways can 2 boys and 3 girls be selected from 5 boys and 7 girls? (2020)

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Question: How many ways can 2 boys and 3 girls be selected from 5 boys and 7 girls? (2020)

Options:

Correct Answer: 350

Exam Year: 2020

Solution:

The number of ways is 5C2 * 7C3 = 10 * 35 = 350.

How many ways can 2 boys and 3 girls be selected from 5 boys and 7 girls? (2020)

How many ways can 2 boys and 3 girls be selected from 5 boys and 7 girls? (2020)

Step 1: Understand that we need to select 2 boys from a group of 5 boys.
Step 2: Use the combination formula to find the number of ways to choose 2 boys from 5. This is written as 5C2.
Step 3: Calculate 5C2. The formula for combinations is nCr = n! / (r! * (n - r)!). Here, n = 5 and r = 2.
Step 4: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 5: Calculate 2! = 2 × 1 = 2.
Step 6: Calculate (5 - 2)! = 3! = 3 × 2 × 1 = 6.
Step 7: Plug the values into the combination formula: 5C2 = 5! / (2! * 3!) = 120 / (2 * 6) = 120 / 12 = 10.
Step 8: Now, understand that we also need to select 3 girls from a group of 7 girls.
Step 9: Use the combination formula to find the number of ways to choose 3 girls from 7. This is written as 7C3.
Step 10: Calculate 7C3 using the same combination formula. Here, n = 7 and r = 3.
Step 11: Calculate 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.
Step 12: Calculate 3! = 3 × 2 × 1 = 6.
Step 13: Calculate (7 - 3)! = 4! = 4 × 3 × 2 × 1 = 24.
Step 14: Plug the values into the combination formula: 7C3 = 7! / (3! * 4!) = 5040 / (6 * 24) = 5040 / 144 = 35.
Step 15: Now, multiply the number of ways to choose the boys and the girls together: 5C2 * 7C3 = 10 * 35.
Step 16: Calculate the final result: 10 * 35 = 350.

Combination – The problem tests the understanding of combinations, specifically how to choose a subset of items from a larger set without regard to the order of selection.
Binomial Coefficient – The use of binomial coefficients (nCr) to calculate the number of ways to choose r items from n items is central to solving this problem.