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In how many ways $20$ mangoes can be divided into $5$ children such that all contains $4$ mangoes to each child.
Option: 1
$4845$

Option: 2
$2546168620$

Option: 3
$2546168625$

Option: 4
$1820$

Answers (1)

best_answer

The given information is:

Number of mangoes $=20$

Number of children $=5$

Number of mangoes to each child $=4$

Pick first $4$ mangoes from $20$ mangoes ,

Total no. of ways of choosing is:

$\frac{20 !}{(20-4) ! 4 !}=\frac{20 !}{16 ! 4 !}=4845$

Pick next $4$ mangoes from $16$ mangoes ,

Total no. of ways of choosing is:

$\frac{16 !}{(16-4) ! 4 !}=\frac{16 !}{12 ! 4 !}=1820$

Pick next $4$ mangoes from $12$ mangoes ,

Total no. of ways of choosing is:

$\frac{12 !}{(12-4) ! 4 !}=\frac{12 !}{8 ! 4 !}=495$

Pick next $4$ mangoes from $8$ mangoes ,

Total no. of ways of choosing is:

$\frac{8 !}{(8-4) ! 4 !}=\frac{8 !}{4 ! 4 !}=70$

Pick last $4$ mangoes from remaining $4$ mangoes and the number of ways this can be done is $1$ .

Since the five groups are similar, there is no differentiation between them. Therefore, we need to divide it with $5 !=120$ .

The total numbers of ways:

$\frac{4845 \times 1820 \times 495 \times 70}{120}=2546168625$

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Riya

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