First slide

Combinations

Question

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

Difficult

A

1500
B

2255
C

3000
D

2250

Solution

$\begin{array}{lcc} A (5) & B (5) & C (5) \\ 1 & 1 & 3 \\ 1 & 3 & 1 \\ 3 & 1 & 1 \\ 2 & 2 & 1 \\ 2 & 1 & 2 \\ 1 & 2 & 2 \end{array}$
Number of ways to choose,
(1,1,3) or (1,3,1) or (3,1,1) questions from A, B, C = $({}^{5}{C_{1} x {}^{5}{C_{1}} x {}^{5}{C_{3}}}) x 3 = 750 w a y s$
(2,2,1) or (2,1,2) or (1,2,2) questions from A, B, C = $({}^{5}{C_{2} x {}^{5}{C_{2}} x {}^{5}{C_{1}}}) x 3 = 1500 w a y s$
Hence, the total number ways = 2250

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