Question

. in how many ways can a football team of 11 players be selected from 16 players? how many of them will (i) include 2 particular players? (ii) exclude 2 particular players?

Answer 1

The total number of ways to select the team is 4368.

(i)the number of ways to include 2 particular players is 2002 and

(ii) the number of ways to exclude 2 particular players is 364

Step-by-step explanation:

We will first determine the number of ways we can select a football team of 11 players from 16 players. This is a combination problem where we are not concerned with the order of selection, only the choice of players.
To compute this, we can use the combination formula, which is:

`\[ C(n, k) = (n!)/(k!(n - k)!) \]`
where:
- C(n, k) denotes the number of combinations of n items taken k at a time.
- n is the total number of items.
- k is the number of items to be chosen.
- n! denotes the factorial of n, which is the product of all positive integers less than or equal to n.
- k! and (n - k)! are the factorials of k and (n - k), respectively.

For our case, (n = 16) (total players) and (k = 11) (team size).
So, the total number of ways to select the team is:

`\[ C(16, 11) = (16!)/(11! \cdot (16 - 11)!)`

`= (16!)/(11! \cdot 5!) \]`

`=4368`


(i) Next, we want to find out in how many of those combinations 2 particular players are included.
If we want to include 2 specific players, then we only need to choose the remaining ( 11 - 2 = 9) players from the remaining ( 16 - 2 = 14) players.
Using the combination formula again:

`\[ C(14, 9) = (14!)/(9! \cdot (14 - 9)!)`

`= (14!)/(9! \cdot 5!) \]`

`=2002`

This is the number of ways to form a team that includes the 2 particular players.

(ii)To find out in how many of those combinations 2 particular players are excluded, we simply choose all 11 players from the remaining 14 players (since we're excluding 2).
Using the combination formula:

`\[ C(14, 11) = (14!)/(11! \cdot (14 - 11)!)`

`= (14!)/(11! \cdot 3!) \]`

=364

Hence, the total number of ways to select the team is 4368, the number of ways to include 2 particular players is 2002 and the number of ways to exclude 2 particular players is 364.