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Question

The number of ways to make 5 heaps of 3books each from 15 different books is-

  1. fraction numerator 15 factorial over denominator 5 factorial left parenthesis 3 factorial right parenthesis to the power of 5 end exponent end fraction    
  2. fraction numerator 15 factorial over denominator left parenthesis 3 factorial right parenthesis to the power of 5 end exponent end fraction    
  3. 15C3    
  4. 15P5    

hintHint:

Here we will be using the concept of arrangement of distinct objects into equal groups.

The correct answer is: fraction numerator 15 factorial over denominator 5 factorial left parenthesis 3 factorial right parenthesis to the power of 5 end exponent end fraction


    Detailed Solution :
    5 heaps of 3 books each are to be made from 15 different books. We are to find in how many ways this can be done.
    W e space k n o w space t h a t space t h e space f o r m u l a space o f space d i v i d i n g space m space space d i f f e r e n t space t h i n g s space i n t o space g r o u p s space o f space s i z e s space a subscript 1 comma a subscript 2 comma a subscript 3... comma a subscript n space space w h e r e

a subscript 1 plus a subscript 2 plus a subscript 3 plus... plus a subscript n equals space fraction numerator m factorial over denominator left parenthesis a subscript 1 factorial right parenthesis left parenthesis a subscript 2 factorial right parenthesis left parenthesis a subscript 3 factorial right parenthesis... left parenthesis a subscript n factorial right parenthesis space. end fraction
    But this is applicable on if the groups are of unequal sizes.
    In the given problem all the groups are of size 3, and there are 5 groups.
    Hence,
    a subscript 1 equals space a subscript 2 equals space a subscript 3 equals space a subscript 4 equals space a subscript 5 equals 3
m equals 15
    Again, the equal groups will be arranged amongst themselves, which is possible in 5! ways. Thus, we can say that the required number of ways in which the 15 different books can be divided into 5 groups of size 3 is
    fraction numerator 15 factorial over denominator left parenthesis 3 factorial right parenthesis left parenthesis 3 factorial right parenthesis left parenthesis 3 factorial right parenthesis left parenthesis 3 factorial right parenthesis left parenthesis 3 factorial right parenthesis times left parenthesis 5 factorial right parenthesis end fraction
    equals space fraction numerator 15 factorial over denominator left parenthesis 3 factorial right parenthesis to the power of 5 times left parenthesis 5 factorial right parenthesis end fraction



     

    In these type of questions, it is to be always remembered that separate approaches need to be adopted for distinct and identical objects. Here, the books were distinct or different.

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