In How Many Ways can 5 Apples (Identical) be Distributed Among 4 Children? GMAT Problem Solving

Question: In how many ways can 5 apples (identical) be distributed among 4 children? (Some children may get no apples.)

(A) 56
(B) 144
(C) 200
(D) 256
(E) 312

Answer: A
Solution and Explanation:

Approach Solution 1:

There are four children, and we have to distribute five candies.
The candidate should keep in mind the following permutation and combination formulas in order to complete the issue.
nPr = n! / (n-r) is a permutation of n objects taken r at a time.
combination of n items, each taken r times: ((n-r)! = nCr = n! * r!)
in which x! = x*(x-1)*(x-2)* (3).
Number of ways of dividing 'n' identical objects into 'r' groups such that each group can contain any number of objects is given by
n+r-1C r-1
So,
When there are 4 kids and 5 apples, the number of possible splits is
(5+4-1) C (4-1)
= 8C3
= 8! / (5!∗3!)
= 56
Hence A is the correct answer.

Approach Solution 2:

There are four children, and we have to distribute 5 candies.
The candidate should keep in mind the following permutation and combination formulas in order to complete the issue.
nPr = n! / (n-r) is a permutation of n objects taken r at a time.
combination of n items, each taken r times: ((n-r)! = nCr = n! * r!)
in which x! = x*(x-1)*(x-2)* (3).
Let's assume that apples are represented by the letter A and that dividing them into four baskets will be enough to distribute the apples to various recipients.
Three partition bars are required to divide the apples into four baskets, as shown below:
Kid 1 _ kid 2 _ kid 3 _ kid 4 : A | A | AA | A
How many different ways can we arrange our three identical partition bars and our five identical apples?
8! / (5!3!) = 56
Hence A is the correct answer.

“In how many ways can 5 apples (identical) be distributed among 4 children?" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics.

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