Question

In a shooting competition, a man can score 0, 2 and 4 points for each shot. Then the number of different ways in which he can score 14 points in 5 shots is N, then the number of digits in N equals?

Answer 1

Hint: For solving these types of questions, you should know about the recursion and permutations and combinations and the probability distributions. We can solve this question by the help of probability distribution.

Complete step-by-step solution:
The question is asking us to find the number of ways in which the man can score 14 points in 5 shots. He gets the points of 0, 2 and 4 for each shot. And he scored 14 points in 5 shots. So, we can find the number of ways in which the man gets those 14 points in just 5 shots as follows:
1. He can get 4, 4, 4, 2, 0 at each time. So, the ways for getting this are,
$\begin{align}
  & \dfrac{5!}{3!}=\dfrac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1} \\
 & =20 \\
\end{align}$
2. He can also get 2, 2, 2, 4, 4 the next time and he can get 14 points in 5 shots. So, the ways for getting 2, 2, 2, 4, 4 are,
$\begin{align}
  & \dfrac{5!}{3!2!}=\dfrac{5\times 4\times 3\times 2\times 1}{3\times 2\times 2\times 1} \\
 & =10 \\
\end{align}$
So, the total number of ways for scoring 14 points in 5 shots are,
$N=20+10=30$
Therefore, the man can get the 14 points with 5 shots with the help of 30 ways.

Note: For solving this type of questions, we have to find the different ways to get the final result and there is a very much use of the factorial to find the number of ways to get the correct answer. Using this, all the ways to get the target can be calculated in a good way.