Question

In how many ways can a football team of 11 players be selected from 15 players. How many of these will
(i) Include one particular player
(ii) Exclude one particular player

Answer 1

Hint: Selection of r different things from n different things is given by the expression
$c_{r}^{n}=\dfrac{n!}{r!\left( n-r \right)!}$. For including one person, include him in the team and select other players from the remaining players. For excluding, remove one person from total players present and select total players required from the remaining players.

Complete step-by-step answer:
As we know the number of ways of selecting ‘r’ things from ‘n’ different objects can be given by relation:
$c_{r}^{n}=\dfrac{n!}{r!\left( n-r \right)!}$ ………………… (i)
So, hence we need to select a football time of 11 players from 15 players. Hence using the number of teams formed with 11 players as $c_{11}^{15}$ .
Now, using the equation (i), we can simplify it as
$c_{11}^{15}=\dfrac{15!}{4!11!}$
As, we can write 15! as $15\times 14\times 13\times 12\times 11!$
So, we get
$\begin{align}
  & c_{11}^{15}=\dfrac{15\times 14\times 13\times 12\times 11!}{4!11!} \\
 & =\dfrac{15\times 14\times 13\times 12}{4\times 3\times 2\times 1}=105\times 13 \\
\end{align}$
= 1365
Hence, there are 1365 ways to form a football team with 11 players from 15 players.
i) Include one particular player:
As it is already given that the football team consists of 11 players. Now, if we want to include one particular player out of 15, that player will always be in 11 selected players. It means that we have selected one particular player, out of the remaining 14 players. So, we can select 10 players out of 14 players with the help of equation (i) as $c_{10}^{14}$. So, we get
$\begin{align}
  & c_{10}^{14}=\dfrac{14!}{10!4!}=\dfrac{14\times 13\times 12\times 11\times 10!}{10!\times 4\times 3\times 2\times 1} \\
 & =91\times 11 \\
\end{align}$
= 1001
Hence, there will be 1001 ways to select 11 players from 15 plays if we want to include one particular player always.
ii) Exclude one particular player:
As, we need to select 11 players for the football time from given 15 players but here we are not considering one particular player for selection as we need to exclude him/her. It means we need to select 11 players from the remaining 14 players.
Hence, it can be given with the help of a relation given in the equation as $c_{11}^{14}$. So, we can solve it as:
$c_{11}^{14}=\dfrac{14!}{3!11!}=\dfrac{14\times 13\times 12\times 11!}{3\times 2\times 1\times 11!}=14\times 26$
= 364
Hence, we can form a football team of 11 players from 15 players by 365 ways, if we are excluding one particular player.

Note: One may go wrong with the cases of excluding and including. Just do the process of exclusion or inclusion at the very starting of the solution, either select 10 players from 15 players or select 11 players from 15 players then try to add or subtract some cases, which will be a very complex approach and he/she may lose some of the cases. So, be careful with these kinds of questions and do exclusion or inclusion at the very starting.
Don’t use the $P_{r}^{n}$ formula for selecting the players. As it will select the players and arrange them as well. So, we need to do only the selection part, not arrangement of the players. Hence, don’t arrange them using another approach as well. Be clear with the concepts of selection and arrangement.