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

Last updated date: 18th Jun 2024
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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 ${}^n{P_r}$.

Thus we can say that the required no of ways = ${}^7{P_3}$ where n=7 and r=3.
${}^7{P_3}{\rm{ = }}\dfrac{{7!}}{{(7 - 3)!}}$
$\begin{array}{l}{\rm{ = }}\dfrac{{7!}}{{4!}}{\rm{ = }}\dfrac{{7{\rm{ x 6 x 5 x 4!}}}}{{4!}}\\{}^7{P_3}{\rm{ = 7 x 6 x 5 = 210}}\end{array}$
Note: If it was asked to find no. of ways in which any three athletes win the prize, then we take combination$\begin{array}{l}{}^n{C_r}{\rm{ = }}\dfrac{{n!}}{{(n - r)!{\rm{ r!}}}}\\{}^7{C_3}{\rm{ = }}\dfrac{{7!}}{{(7 - 3)!{\rm{ 3!}}}}{\rm{ = }}\dfrac{{7!}}{{4!3!}}{\rm{ = }}\dfrac{{7{\rm{ x 6 x 5}}}}{{3{\rm{ x 2}}}}{\rm{ = 35 ways}}{\rm{.}}\end{array}$