A team of 5 children is to be selected out of 4 girls and 5 boys such that it contains at least 2 girls. In how many different ways the selection can be made?
Correct Answer: Description for Correct answer:
A team of 5 children consisting of at least two girls can be formed in following ways:
I. Selecting 2 girls out of 4 and 3 boys out of 5. This can be done in \( \Large ^{4}C_{2} \times ^{5}C_{3} \) ways. '
ll. Selecting, 3 girls out of 4 and 3 boys out of 5. This can be done in \( \Large ^{4}C_{3} \times ^{5}C_{2} \) ways.
III. Selecting 4 girls out of 4 and 1 boy out of 5. This can be done in \( \Large ^{4}C_{4} \times ^{5}C_{1} \) ways.
Since the team is formed in each case, therefore, by the fundamental principle of addition, the total number of ways of forming the team.
\( \Large ^{4}C_{3} \times ^{5}C_{3} \)
=\( \Large ^{4}C_{2} \times ^{5}C_{3} + ^{4}C_{3} \times ^{5}C_{2} + ^{4}C_{4} \times ^{5}C_{1} \)
=\( \Large \frac{4 \times 3}{1 \times 2} \times \frac{5 \times 4 \times 3}{1 \times 2 \times 3} + \frac{4 \times 3 \times 2}{1 \times 2 \times 3} \times \frac{5 \times 4}{1 \times 2} + 1 \times 5 \)
= 60 + 40 + 5 = 105
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