A student has to answer 10 question, choosing at least 4 from each of

1.	A student has to answer 10 question, choosing at least 4 from each of part A and B. If there are 6 questions in part A and 7 in part B. In how many ways can the student choose 10 questons?
Answer» Combination from A and from B ∷ \(\frac{4}{5};\frac{5}{5};\frac{6}{4}\) Number of way to get \(\frac{4}{6}\) pattern = ⁶C₄ × ⁷C₆ = \(\frac{6!}{2!\space4!}\times\frac{7!}{1!\space6!}\) = \(\frac{6.5}{2.1}\times\frac{7}{1}\) = 15 × 7 = 105 Number of ways to get \(\frac{5}{5}\)pattern = ⁶C₅ × ⁷C₅ = \(\frac{6!}{1!\space5!}\times\frac{7!}{2!\space5!}\) = \(\frac{6}{1}\times\frac{7\times6}{2\times1}\) = 6 × 21 = 126 Number of ways to get the 6 4 pattern = ⁶C₆ × ⁷C₄ = \(\frac{6!}{0!\space 6!}\times\frac{7!}{3!\space4!}\) = \(1\times\frac{7.6.5}{3.2.1}\) = 35 Hence, total number of ways= 105 + 126 + 35 = 266

A student has to answer 10 question, choosing at least 4 from each of part A and B. If there are 6 questions in part A and 7 in part B. In how many ways can the student choose 10 questons?

Answer»

Combination from A and from B ∷ \(\frac{4}{5};\frac{5}{5};\frac{6}{4}\)

Number of way to get \(\frac{4}{6}\) pattern = ⁶C₄ × ⁷C₆

= \(\frac{6!}{2!\space4!}\times\frac{7!}{1!\space6!}\)

= \(\frac{6.5}{2.1}\times\frac{7}{1}\)

= 15 × 7 = 105

Number of ways to get \(\frac{5}{5}\)pattern = ⁶C₅ × ⁷C₅

= \(\frac{6!}{1!\space5!}\times\frac{7!}{2!\space5!}\)

= \(\frac{6}{1}\times\frac{7\times6}{2\times1}\)

= 6 × 21 = 126

Number of ways to get the 6 4 pattern = ⁶C₆ × ⁷C₄

= \(\frac{6!}{0!\space 6!}\times\frac{7!}{3!\space4!}\)

= \(1\times\frac{7.6.5}{3.2.1}\)

= 35

Hence, total number of ways= 105 + 126 + 35

= 266