A committee of ten is to be formed from eight teachers and twelve students of whom four are girls. The number of committees which contains atleast four of either group (teachers and students) and atleast two girls and atleast two boys is equal to m . The number of divisors of m is n , then n 6 is equal to

Question

A committee of ten is to be formed from eight teachers and twelve students of whom four are girls. The number of committees which contains atleast four of either group (teachers and students) and atleast two girls and atleast two boys is equal to m . The number of divisors of m is n , then n 6 is equal to,

MARKS App · Accepted Answer

Teachers 8 Students 12 8 B + 4 G 4 6 2 B 4 G , 3 B 3 G , 4 B 2 G 5 5 2 B 3 G , 3 B 2 G 6 4 2 B , 2 G Required number of ways = m = C 8 4 C 8 2 C 4 4 + C 8 3 C 4 3 + C 8 4 C 4 2 + C 8 5 C 8 2 C 4 3 + C 8 3 C 4 2 + C 8 6 C 8 2 C 4 2 = 70 28 + 56 × 4 + 70 × 6 + 56 28 × 4 + 56 × 6 + 28 28 × 6 = 47040 + 25088 + 4704 ⇒ m = 76832 m = 2 5 ⋅ 7 4 Number of divisions n = 5 + 1 4 + 1 = 30 So, n 6 = 5

Teachers $(8)$	Students $(12)$	$8 B + 4 G$

$4$	$6$	$2 B 4 G, 3 B 3 G, 4 B 2 G$
$5$	$5$	$2 B 3 G, 3 B 2 G$
$6$	$4$	$2 B, 2 G$

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