There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen 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 batsmen and 1 wicketkeeper, is ___.

We have 6 bowlers (B), 7 batsmen (bat), and 2 wicketkeepers (wk), and we need to select 11 players with at least 4 bowlers, at least 5 batsmen, and at least 1 wicketkeeper.

The valid combinations $$(B, \text{bat}, \text{wk})$$ that satisfy all constraints and sum to 11 are:

Case 1: $$(4, 5, 2)$$: $$\binom{6}{4}\binom{7}{5}\binom{2}{2} = 15 \times 21 \times 1 = 315$$

Case 2: $$(4, 6, 1)$$: $$\binom{6}{4}\binom{7}{6}\binom{2}{1} = 15 \times 7 \times 2 = 210$$

Case 3: $$(5, 5, 1)$$: $$\binom{6}{5}\binom{7}{5}\binom{2}{1} = 6 \times 21 \times 2 = 252$$

Cases with 0 wicketkeepers or fewer than 5 batsmen are excluded by the constraints. The total number of ways is $$315 + 210 + 252 = 777$$.