Question

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Hint:

Use the combination formula to determine the number of ways to select items from a set.

Answer 1

Step 1: Identify the number of balls to be selected.
We need to select 2 black balls from 5 and 3 red balls from 6.
Step 2: Apply the combination formula.
The number of ways to choose \( r \) items from \( n \) items is given by the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Step 3: Calculate the number of ways to select the black balls.
The number of ways to select 2 black balls from 5 is: \[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
Step 4: Calculate the number of ways to select the red balls.
The number of ways to select 3 red balls from 6 is: \[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
Step 5: Multiply the two values to get the total number of ways.
The total number of ways to select 2 black and 3 red balls is: \[ \text{Total ways} = C(5, 2) \times C(6, 3) = 10 \times 20 = 200 \]