Question 15 Marks
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
Answer
View full question & answer→Total number of quation = 12
Total number of quatin to be answered = 7
Each group has 6 quation more than 5 quation from either grou is not permitted, the number of ways a student can choose quation,
${^{6}{\text{C}}}_{\text{2}}\times{^{6}{\text{C}}}_{\text{5}}+{^{6}{\text{C}}}_{\text{4}}\times{^{6}{\text{C}}}_{\text{4}}+{^{6}{\text{C}}}_{\text{4}}\times{^{6}{\text{C}}}_{\text{3}}+{^{6}{\text{C}}}_{\text{5}}\times{^{6}{\text{C}}}_{\text{2}}$
$=2\Big({^{6}{\text{C}}}_{\text{2}}\times{^{6}{\text{C}}}_{\text{5}}+{^{6}{\text{C}}}_{\text{3}}\times{^{6}{\text{C}}}_{\text{4}}\Big)$
$=2\Big(\frac{6!}{2!4!}\times\frac{6!}{5!1!}+\frac{6!}{3!3!}\times\frac{6!}{4!2!}\Big)$
$=\frac{2\times6\times5\times6}{2}\Big(1+\frac{20}{6}\Big)$
$=30\times26=780$
Total number of quatin to be answered = 7
Each group has 6 quation more than 5 quation from either grou is not permitted, the number of ways a student can choose quation,
${^{6}{\text{C}}}_{\text{2}}\times{^{6}{\text{C}}}_{\text{5}}+{^{6}{\text{C}}}_{\text{4}}\times{^{6}{\text{C}}}_{\text{4}}+{^{6}{\text{C}}}_{\text{4}}\times{^{6}{\text{C}}}_{\text{3}}+{^{6}{\text{C}}}_{\text{5}}\times{^{6}{\text{C}}}_{\text{2}}$
$=2\Big({^{6}{\text{C}}}_{\text{2}}\times{^{6}{\text{C}}}_{\text{5}}+{^{6}{\text{C}}}_{\text{3}}\times{^{6}{\text{C}}}_{\text{4}}\Big)$
$=2\Big(\frac{6!}{2!4!}\times\frac{6!}{5!1!}+\frac{6!}{3!3!}\times\frac{6!}{4!2!}\Big)$
$=\frac{2\times6\times5\times6}{2}\Big(1+\frac{20}{6}\Big)$
$=30\times26=780$