The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:
The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:
- 205
- 615
- 510
- 430
The Correct Option is D
Solution and Explanation
The correct answer is (D) : 430
By multinomial theorem, no. of ways to distribute 30 identical candies among four children C1, C2 and C3, C4
= Coefficient of x30 in (x4 + x5 + … + x7) (x2 + x3 +…+ x6) (1 + x + x2…)2
=Coefficient of x24 in \(\frac{(1−x^4)(1−x^5)(1−x^{31})^2}{(1−x)(1-x)(1-x)^2}\)
= Coefficient of x24 in (1 – x4 – x5 + x9) (1 – x)–4
= 27C24 – 23C20 – 22C19 + 18C15 = 430
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.