Question

find the number of ways in which 18 identical balls can be used in 15 different cricket matches.

Answer 1

$\binom{32}{14}$

Answer 2

The problem asks us to find the number of ways to distribute 18 identical balls into 15 different cricket matches. This is a classic problem of distributing identical items into distinct bins, which can be solved using the stars and bars method.

Let $x_i$ be the number of balls used in the $i$-th cricket match, where $i$ ranges from 1 to 15. Since the balls are identical and the matches are different, we are looking for the number of non-negative integer solutions to the equation:

$x_1 + x_2 + \dots + x_{15} = 18$

Here, $n = 18$ (the total number of identical items, i.e., balls) and $k = 15$ (the number of distinct bins, i.e., cricket matches). Since no minimum number of balls per match is specified, each $x_i$ can be zero or any positive integer ($x_i \ge 0$).

The number of non-negative integer solutions to an equation of the form $x_1 + x_2 + \dots + x_k = n$ is given by the formula:

$$\binom{n+k-1}{k-1} \quad \text{or equivalently} \quad \binom{n+k-1}{n}$$

Substituting the values $n=18$ and $k=15$ into the formula:

Number of ways = $\binom{18 + 15 - 1}{15 - 1}$

Number of ways = $\binom{32}{14}$

Alternatively, using the second form of the formula:

Number of ways = $\binom{18 + 15 - 1}{18}$

Number of ways = $\binom{32}{18}$

Both expressions are equivalent since $\binom{N}{R} = \binom{N}{N-R}$.

The number of ways is $\binom{32}{14}$.