Question

In how many ways can a football team of 11 players be selected from 16 players. So that a particular player is included?

Answer 1

Hint: Find the number of ways in which 11 players can be selected from 16 players by using combination. Now we need to include one particular player. thus find the no of ways of selection of 10 players from 15 as one is already fixed. Select 11 players.

Complete step-by-step answer:
Now the 11 players out of 16 players can be selected by the process of combination.
$\therefore$11 players can be selected out of 16 in $^{16}{{C}_{11}}$ ways.
Now this is of the form $^{n}{{C}_{r}}=\dfrac{n!}{(n-r)!r!}$.
${{\therefore }^{16}}{{C}_{11}}=\dfrac{16!}{(16-11)!11!}=\dfrac{16\times 15\times 14\times 13\times 12\times 11!}{5!\times 11!}$
$=\dfrac{16\times 15\times 14\times 13\times 12}{5\times 4\times 3\times 2\times 1}=4\times 3\times 7\times 13\times 4=4368$ ways.
Now we need to include a particular player. If one particular player has to be included then the selection will be $^{15}{{C}_{10}}$ ways.

& {{\therefore }^{15}}{{C}_{10}}=\dfrac{15!}{(15-10)!10!}=\dfrac{15\times 14\times 13\times 12\times 11\times 10!}{5!\times 10!} \\\ & =\dfrac{15\times 14\times 13\times 12\times 11}{5\times 4\times 3\times 2\times 1}=3\times 7\times 13\times 11=3003. \\\ \end{aligned}$$ Thus if one particular player is to be included in selection, it will be in 3003 ways. Note: If we need to add 2 players, then the number of ways would be $$^{14}{{C}_{9}}=2002$$ ways. If we need to exclude a particular player, then it can be done out of the total players in $$^{15}{{C}_{11}}=1365$$ ways.