MHT CET · Maths · Permutation Combination
A linguistic club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this group including the selection of a leader (from among these 4 members) for the team. If the team has to include at most one boy, the number of ways of selecting the team is
- A \(140\)
- B \(320\)
- C \(76\)
- D \(380\)
Answer & Solution
Correct Answer
(D) \(380\)
Step-by-step Solution
Detailed explanation
Case I: No boy is included.
Selecting 4 girls from 6 girls \(={ }^6 \mathrm{C}_4\)
Selecting 1 captain from selected members \(={ }^4 \mathrm{C}_1\)
Total number of ways \(={ }^6 \mathrm{C}_4 \times{ }^4 \mathrm{C}_1=60\)
Case II: One boy is included.
Selecting 3 girls and 1 boy from given members \(={ }^6 \mathrm{C}_3 \times{ }^4 \mathrm{C}_1\).
Selecting 1 captain from the selected members \(={ }^4 \mathrm{C}_1\).
Total Number of ways \(={ }^6 \mathrm{C}_3 \times{ }^4 \mathrm{C}_1 \times{ }^4 \mathrm{C}_1=320\).
\(\therefore \) Total Number of ways \(=320+60=380\).
Selecting 4 girls from 6 girls \(={ }^6 \mathrm{C}_4\)
Selecting 1 captain from selected members \(={ }^4 \mathrm{C}_1\)
Total number of ways \(={ }^6 \mathrm{C}_4 \times{ }^4 \mathrm{C}_1=60\)
Case II: One boy is included.
Selecting 3 girls and 1 boy from given members \(={ }^6 \mathrm{C}_3 \times{ }^4 \mathrm{C}_1\).
Selecting 1 captain from the selected members \(={ }^4 \mathrm{C}_1\).
Total Number of ways \(={ }^6 \mathrm{C}_3 \times{ }^4 \mathrm{C}_1 \times{ }^4 \mathrm{C}_1=320\).
\(\therefore \) Total Number of ways \(=320+60=380\).
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