# Find the number of ways of selecting 9 balls from 6 red balls,

Question:

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Solution:

There are a total of 6 red balls, 5 white balls, and 5 blue balls.

9 balls have to be selected in such a way that each selection consists of 3 balls of each colour.

Here,

3 balls can be selected from 6 red balls in ${ }^{6} \mathrm{C}_{3}$ ways.

3 balls can be selected from 5 white balls in ${ }^{5} \mathrm{C}_{3}$ ways.

3 balls can be selected from 5 blue balls in ${ }^{5} \mathrm{C}_{3}$ ways.

Thus, by multiplication principle, required number of ways of selecting 9 balls

$={ }^{6} \mathrm{C}_{3} \times{ }^{5} \mathrm{C}_{3} \times{ }^{5} \mathrm{C}_{3}=\frac{6 !}{3 ! 3 !} \times \frac{5 !}{3 ! 2 !} \times \frac{5 !}{3 ! 2 !}$

$=\frac{6 \times 5 \times 4 \times 3 !}{3 ! \times 3 \times 2} \times \frac{5 \times 4 \times 3 !}{3 ! \times 2 \times 1} \times \frac{5 \times 4 \times 3 !}{3 ! \times 2 \times 1}$

$=20 \times 10 \times 10=2000$