A linguistic club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this group…
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
$140$
$320$
$76$
$380$
Solution
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 $={ }^6 \mathrm{C}_3 \times{ }^4 \mathrm{C}_1 \times{ }^4 \mathrm{C}_1=320$.
$\therefore \quad$ Total Number of ways $=320+60=380$.