There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ______________.

Solution

Maths sir · 3 steps

The minimums already fill 10 places, so just ask which group the 11th player comes from.

1. One spare place. At least 4 bowlers, 5 batsmen and 1 wicketkeeper make 4 + 5 + 1 = 10 players. The team has 11, so exactly one extra player is added. That player is a bowler, a batsman or a wicketkeeper, which gives three separate cases.

   Bowlers, batsmen, keepers	Ways	Count
   5, 5, 1	C(6, 5)C(7, 5)C(2, 1) = 6 × 21 × 2	252
   4, 6, 1	C(6, 4)C(7, 6)C(2, 1) = 15 × 7 × 2	210
   4, 5, 2	C(6, 4)C(7, 5)C(2, 2) = 15 × 21 × 1	315

2. Add the cases. No team can fall into two cases, because each case has different group sizes. So the case totals add directly: 252 + 210 + 315.
3. So. The number of ways is 777.

The common slip: 3150. Picking the minimum 10 first and then any 1 of the remaining 5 gives 15 × 21 × 2 × 5 = 3150. That counts the same team several times, so split into cases instead.

The idea to keep. For an 'at least' selection, first fill the minimums, then split into cases by where the extra places go. Count each case with a product of combinations and add the cases.

Answer from the NTA final answer key. Solution by PrepWiser.