Question

Find the number of ways in which five bowlers of different ages can choose to bowl on over each one after the other, if the oldest bowler should not bowl the first over.

Answer 1

Here, we need to find the number of ways in which five bowlers of different ages can choose to bowl, if the oldest bowler should not bowl the first over. We will use the given information to obtain the choices available to each over and then use product rule to obtain the final answer. The product rule states that if there are $a$ number of ways to do the task $A$ and $b$ number of ways to do task $B$, then the total number of ways can be counted by $a \times b$.

Complete step by step solution:
Let the bowlers be A, B, C, D, and E, where E is the oldest bowler.
We know that the first over cannot be bowled on by the oldest bowler E.
Thus, the first over can be bowled by A, B, C, or D.
Therefore, there are 4 ways to select the bowler who bowls the first over.
Now, the second over can be bowled by anyone except the bowler who bowled the first over. For example, if A bowls the first over, B, C, D, or E can bowl the second over.
Therefore, there are 4 ways to select the bowler who bowls the second over.
The third over can be bowled by anyone except the bowlers who bowled the first two overs. For example, if A and B bowled the first two overs, C, D, or E can bowl the third over.
Therefore, there are 3 ways to select the bowler who bowls the third over.
Similarly, the fourth over can be bowled by anyone except the bowlers who bowled the first three overs. For example, if A, B and C bowled the first three overs, D or E can bowl the fourth over.
Therefore, there are 2 ways to select the bowler who bowls the fourth over.
Finally, the fifth over can be bowled by anyone except the bowlers who bowled the first four overs. For example, if A, B, C, and D bowled the first four overs, E will bowl the fifth over.
Therefore, there is 1 way to select the bowler who bowls the fifth over.
We can find the total number of ways as
${\text{total number of ways}} = 4 \times 4 \times 3 \times 2 \times 1 = 96$

Thus, there are 96 ways in which five bowlers of different ages can choose to bowl on over each one after the other, if the oldest bowler should not bowl the first over.

Note:
We can also solve this problem using the subtraction rule.
We will assume that any bowler can bowl the first over.
The number of ways to select the bowlers who bowl the 5 overs is $5 \times 4 \times 3 \times 2 \times 1 = 120$.
Now we will assume that the oldest bowler bowls the first bowler.
The number of ways to select the bowlers who bowl the next 4 overs is $4 \times 3 \times 2 \times 1 = 24$.
The number of ways to select the bowlers in which the first over is not bowled by the oldest bowler is the difference in the number of ways to select the bowlers who bowl the 5 overs (if anyone can bowl the first over), and the number of ways to select the bowlers who bowl the next 4 overs if the oldest bowler bowls the first over.
Thus, we get the answer as $120 - 24 = 96$ ways.