Question

Number of ways in which 5 identical objects can be distributed in 8 persons such that no person gets more than one objects is
$A)8$
$B{)^8}{C_5}$
$C{)^8}{P_5}$
$D)$ None of these

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
Since there are a total of eight persons and we need to find the number of ways in which five identical objects (can be anything) is distributed to that eight persons with a restriction is no person will get more than one objects like If person A gets the object then he will not get any other objects in rest of the objects.
Formula used: ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$(combination)

Complete step-by-step solution:
Let the total objects in five but we need to give that to eight persons, so the possibility of one person getting the object is zero (may not get that object) or one. We are going to find this problem using a combination formula which is ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ where n is the total number of ways and r is the possibility of the combination of the outcomes in that number of ways, hence using this formula we can find the number of ways that the object distributing $5$ identical objects into the total $8$ person with a restriction that none of the people will get more that one object is $^8{C_5}$ (eight is the total person and five is the number of ways of getting the outcome) thus we get option $B{)^8}{C_5}$ is correct.
Since for option $A)8$ (outcome cannot be eight because total objects are five only) hence it is wrong
Also, the option $C{)^8}{P_5}$ is wrong because (permutation of the person, is the total arrangements not the total number of ways) hence all other options are eliminated.

Note: Since the question is about the number of ways so we used a combination formula; if the question is about the total number of arrangements, then we must use another formula which is a permutation ${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$ (n is the total number and r is the number of arrangements).