Question

A swimming team has eight members, only two of which are boys. The coach wants to take a delegation from the team to a special swimming camp. If the delegation must have either five or six members, and must include atleast one boy, then the number of ways to select the delegation, is

• A. 66
• B. 77
• C. 88
• D. 99

Answer 1

Answer: (B)

METHOD - 1
CASE - 1 : Delegation consists of $5$ members
(a) $1$ Boy $+4$ Girls ${ }^{2} C _{1} \times{ }^{6} C _{4}=30$
(b) $2$ Boys $+3$ Girls ${ }^{2} C _{2} \times{ }^{6} C _{3}=20$
$50$
CASE - 2 : Delegation consists of $6$ members
(a) $1$ Boy $+5$ Girls ${ }^{2} C _{1} \times{ }^{6} C _{5}=12$
(b) $2$ Boys $+4$ Girls $2 C 2 \times{ }^{6} C _{4}=15$
$77$ ways
METHOD-2
CASE - 1 : Delegation consists of $5$ members
Total number of ways of selecting $5$ member team number of ways when no boy is selected
$={ }^{8} C _{5}-{ }^{6} C _{5}=50$
CASE - 2 : Delegation consists of 6 members
$\therefore $ Total - no, boy is selected
$={ }^{8} C _{6}-{ }^{6} C _{6}=27 $
Total $=50 +27=77$