Question

The number of ways in which $7$ distinct objects can be distributed among $4$ children is?
(A) $^{\text{7}}{{\text{P}}_{\text{4}}}$
(B) $7!$
(C) $4!$
(D) None of these

Answer 1

In the given question, we have to find the number of ways in which a certain number of distinct objects can be distributed among four children. So, we will first find the number of ways of selecting the number of objects to be given to each child. Then, we find the number of arrangements of these distinct objects. Then, we will use the multiplication rule of counting by multiplying the number of options with each person to find the total number of ways in which $7$ distinct objects can be distributed among $4$ children.

Complete step-by-step solution:
So, we are to find the number of ways in which $7$ distinct objects can be distributed among $4$ children.
So, we know that the number of ways of selecting r objects out of n different objects is $^n{C_r}$. So, we can find the number of ways of selecting the objects to be distributed to each child using the same expression.
So, there are $7$ distinct objects and $4$ children. So, the number of ways of selecting the number of objects to be given to each child is $^7{C_4}$.
Now, we also have to arrange the children as the objects that they are getting are distinct.
We know that the number of arrangements of n things is $n!$.
So, the number of ways in which $4$ children can be arranged is $4!$.
Now, according to the multiplication rule of the counting, if there are m ways of doing one thing and n ways of doing another thing, then there are a total $m \times n$ ways of doing both the things simultaneously.
So, total number of ways in which $7$ distinct objects can be distributed among $4$ children ${ = ^7}{C_4} \times 4!$
Now, we will simplify the expression using the combination formula $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$.
$= \dfrac{{7!}}{{3! \times 4!}} \times 4!$
Cancelling the common factors in numerator and denominator, we get,
$= \dfrac{{7!}}{{3!}}$
Now, we also know that the formula for permutation is as $^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$. So, we can see that $^7{P_4} = \dfrac{{7!}}{{3!}}$.
Hence, we get,
${ = ^7}{P_4}$
So, there are a total of $^{\text{7}}{{\text{P}}_{\text{4}}}$ ways in which $7$ distinct objects can be distributed among $4$ children
So, option (A) is the correct answer.

Note: In such a problem, one must know the multiplication rule of counting as consequent events are occurring. One must take care while doing the calculations and should recheck them so as to be sure of the final answer. We should always first select the objects using the combination formula and then arrange them by multiplying the factorial of the number of objects as the formula of permutation does not work in every situation.