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How many different ways can $20$ colours be divided into $4$ groups of $5$ colour for each group.
Option: 1
$48886$

Option: 2
$488864376$

Option: 3
$4888643$

Option: 4
$1401400$

Answers (1)

best_answer

The given information is:

Number of colour $=20$

Number of group $=4$

Number of colour in each group $=5$

Pick first $5$ colour from $20$ colour,

Total no. of ways of choosing is:

$\frac{20 !}{(20-5) ! 5 !}=\frac{20 !}{15 ! 5 !}=15504$

Pick next $5$ colour from $15$ colour,

Total no. of ways of choosing is:

$\frac{15 !}{(15-5) ! 5 !}=\frac{15 !}{10 ! 5 !}=3003$

Pick next $5$ colour from $10$ colour,

Total no. of ways of choosing is:

$\frac{10 !}{(10-5) ! 5 !}=\frac{10 !}{5 ! 5 !}=252$

Pick last $5$ colour from remaining $5$ colour and the number of ways this can be done is $1$ .

Since the four groups are similar, there is no differentiation between them. Therefore, we need to divide it with $4 !=24$ .

The total numbers of ways :

$\frac{15504 \times 3003 \times 252}{24}=488864376$

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