Question

The number of ways in which first, the second and third prize can be distributed among $5$competitors is? (no person can get more than a prize).
(A) $10$
(B) $15$
(C) $60$
(D) $125$

Answer 1

we are given a problem in which they have to distribute different prizes to the different candidates so there are different ways to distribute the prizes. In this, we have to distribute the prize in order so when the order does matter then we will use permutation.

Formula used:
The number of permutations of n objects taken r at a time is determined by the following formula:
$P(n,r) = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Here $n$= total no of different elements
$r$ = arrangement pattern of the elements
Both $r$ and $n$ are positive integers

Complete step by step answer:
Step1: We are given first, second and third prizes to be distributed among $5$people and one person should not get two prizes. Hence here we have to distribute the prizes in an order
Hence we will use permutations because here order matters
Step2: We will use the formula of permutation
$P(n,r) = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Here $n = 5$ and $r = 3$
Step3: Substituting the value of n and r in the formula we will get the number of permutation
$\Rightarrow {}^5{P_3} = \dfrac{{5!}}{{\left( {5 - 3} \right)!}}$
On expanding the expressions we get
$\Rightarrow {}^5{P_3} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}$
On dividing the numerator by$2$we get:
$\Rightarrow {}^5{P_3} = 5 \times 4 \times 3$
$\Rightarrow {}^5{P_3} = 60$

Hence, the number of ways in which the three prizes can be distributed $= 60$ ways.

Note:
In such types of problems students mainly get confused whether to apply the formula of permutation or of combination. So they have to understand the difference between permutation and combination when order doesn’t matter it is combination and when order matters it is the permutation
We can also solve it simply:
The first prize can be given in $5$ ways. Then the second prize can be given in $4$ ways and the third prize in $3$ ways
(Since a competitor cannot get two prizes) and hence the no. of ways $= 5 \times 4 \times 3$
$= 60$ ways
Keep in mind that
$P(n,r) = \dfrac{{n!}}{{\left( {n - r} \right)!}}$