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How many different ways can $15$ candy be divided into $5$ groups of $3$ candy for each group.
Option: 1
$140$

Option: 2
$1400$

Option: 3
$1401420$

Option: 4
$1401400$

Answers (1)

best_answer

The given information is:

Number of candy $=15$

Number of groups $=5$

Number of candy in each group $=3$

Pick first $3$ candy from $15$ candy,

Total no. of ways of choosing is:

$\frac{15 !}{(15-3) ! 3 !}=\frac{15 !}{12 ! 3 !}=455$

Pick next $3$ candy from $12$ candy,

Total no. of ways of choosing is:

$\frac{12 !}{(12-3) ! 3 !}=\frac{12 !}{9 ! 3 !}=220$

Pick next $3$ candy from $9$ candy,

Total no. of ways of choosing is:

$\frac{9 !}{(9-3) ! 3 !}=\frac{9 !}{6 ! 3 !}=84$

Pick next $3$ candy from $6$ candy,

Total no. of ways of choosing is:

$\frac{6 !}{(6-3) ! 3 !}=\frac{6 !}{3 ! 3 !}=20$

Pick last $3$ candy from $3$ candy 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:

$\frac{455 \times 220 \times 84 \times 20}{120}=1401400$

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