Question

• A. The quantity on the left is greater
• B. The quantity on the right is greater
• C. Both are equal
• D. The relationship cannot be determined without further information

Hint:

Always remember the combination identity \(C(n, r) = C(n, n-r)\). It can save you from calculation and help you quickly see that two seemingly different problems are actually the same, as in this question.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem involves selecting a group of people from two different pools (boys and girls). Since the order of selection for the game does not matter, this is a combination problem. The total number of ways is the product of the ways to select boys and the ways to select girls.
Step 2: Key Formula or Approach:
The number of ways to choose \(r\) items from a set of \(n\) is given by \(C(n, r) = \frac{n!}{r!(n-r)!}\).
An important identity is \(C(n, r) = C(n, n-r)\).
Total Ways = (Ways to choose boys) \( \times \) (Ways to choose girls).
Step 3: Detailed Explanation:
For Column A:
- Select 3 boys from 8. \[ \text{Ways for boys} = C(8, 3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \] - Select 3 girls from 8. \[ \text{Ways for girls} = C(8, 3) = 56 \] - Total number of ways: \[ \text{Total} = 56 \times 56 = 3136 \] So, Quantity A is 3136.
For Column B:
- Select 5 boys from 8. \[ \text{Ways for boys} = C(8, 5) \] Using the identity \(C(n, r) = C(n, n-r)\): \[ C(8, 5) = C(8, 8-5) = C(8, 3) = 56 \] - Select 5 girls from 8. \[ \text{Ways for girls} = C(8, 5) = C(8, 3) = 56 \] - Total number of ways: \[ \text{Total} = 56 \times 56 = 3136 \] So, Quantity B is 3136.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 3136
Quantity B = 3136
The two quantities are equal.