There are fifteen players for a cricket match In how many ways the 11 players can be selected?

Answer

To solve the problem of selecting 11 players from a total of 15 players, we can use the concept of combinations. The number of ways to choose \( r \) objects from \( n \) objects is given by the formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In this case, we need to select 11 players from 15, which can be expressed as:

\[ \binom{15}{11} \]

Using the property of combinations, we know that:

\[ \binom{n}{r} = \binom{n}{n-r} \]

Thus, we can rewrite:

\[ \binom{15}{11} = \binom{15}{4} \]

Now, we can calculate \( \binom{15}{4} \):

\[ \binom{15}{4} = \frac{15!}{4!(15-4)!} = \frac{15!}{4! \cdot 11!} \]

Next, we can simplify this expression. The factorial \( 15! \) can be expanded as:

\[ 15! = 15 \times 14 \times 13 \times 12 \times 11! \]

Substituting this back into our combination formula gives:

\[ \binom{15}{4} = \frac{15 \times 14 \times 13 \times 12 \times 11!}{4! \cdot 11!} \]

The \( 11! \) in the numerator and denominator cancels out:

\[ \binom{15}{4} = \frac{15 \times 14 \times 13 \times 12}{4!} \]

Now, we need to calculate \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Now substituting this back into our equation:

\[ \binom{15}{4} = \frac{15 \times 14 \times 13 \times 12}{24} \]

Next, we can perform the multiplication in the numerator:

\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]

Now, we divide this by 24:

\[ \frac{32760}{24} = 1365 \]

Thus, the number of ways to select 11 players from 15 players is:

\[ \boxed{1365} \]