Question

Seven athletes are participating in a race In how many ways can the first three athletes win the prices?

Answer 1

Hint: The total no. of ways for first 3 prizes can be won is the no. of arrangements of seven athletes to 3 prizes at a time, which is ordered pair is, take the permutation \[{}^n{P_r}\].

Complete step-by-step answer:
It is given that 7 athletes are participating in a race. Thus we can say that total no. of participants = 7.
Now we need to find the no. of ways the first three athletes win the prize. The total no. of ways in which first three prizes can be won is the number of arrangements of seven different things taken 3 at a time. We can solve this using permutation.
Thus we can say that the required no of ways = \[{}^7{P_3}\] where n=7 and r=3.

Thus by applying the formula of permutation wean say,
\[{}^7{P_3}{\rm{ = }}\dfrac{{7!}}{{(7 - 3)!}}\]
\[\begin{array}{l}{\rm{ = }}\dfrac{{7!}}{{4!}}{\rm{ = }}\dfrac{{7{\rm{ x 6 x 5 x 4!}}}}{{4!}}\\{}^7{P_3}{\rm{ = 7 x 6 x 5 = 210}}\end{array}\]

Hence, there are 210 ways in which the first three athletes win the prize.

Note: If it was asked to find no. of ways in which any three athletes win the prize, then we take combination\[\begin{array}{l}{}^n{C_r}{\rm{ = }}\dfrac{{n!}}{{(n - r)!{\rm{ r!}}}}\\{}^7{C_3}{\rm{ = }}\dfrac{{7!}}{{(7 - 3)!{\rm{ 3!}}}}{\rm{ = }}\dfrac{{7!}}{{4!3!}}{\rm{ = }}\dfrac{{7{\rm{ x 6 x 5}}}}{{3{\rm{ x 2}}}}{\rm{ = 35 ways}}{\rm{.}}\end{array}\]