A candidate is required to answer 7 out of 15 questions which are divided into three groups A, B, C each containing 4, 5, 6 questions respectively. He is required to select at least 2 questions from each group. In how many ways can he make up his choice?

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Solution

Only 7 but at least 2 from each group. Hence he has the following choice
(2 A, 2 B, 3C] + (2 A, 3B,2C) + (3A, 2B, 2C]
i.e.( $^{4} C_{2}$ . $^{5} C_{2}$ . $^{6} C_{3}$ ) + ( $^{4} C_{2}$ . $^{5} C_{3}$ . $^{6} C_{2}$ ) + ( $^{4} C_{3}$ . $^{5} C_{2}$ . $^{6} C_{2}$ )
= 1200 + 900 + 600 = 2700

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