First slide

Combinations

Question

Five balls are to be placed in three boxes. Each box can hold all the five and no box remains empty.

Moderate

Question

The number of ways if balls are different but boxes are identical is

A

30
B

25
C

21
D

35

Solution

If no box remains empty, then we can have (1, 1, 3) or (1,2,2) distribution pattern.
When balls are different and boxes are identical, number of distributions is equal to number of divisions in (1, 1, 3) or (1, 2, 2) ways. Hence, total number of ways is
$\frac{5!}{(1!)^{2} 3! 2!} + \frac{5!}{1! (2!)^{2} 2!} = 25$

Question

The number of ways if balls and boxes are identical is

A

3
B

1
C

2
D

none of these

Solution

If no box remains empty, then we can have (1, 1, 3) or (1,2,2) distribution pattern.
When balls as well as boxes are identical, we have only two ways (1, 1, 3) and (1, 2, 2). Hence of ways is 2.

Question

The number of ways if balls as well as boxes are identical but boxes are kept in a row is

A

10
B

15
C

20
D

6

Solution

If no box remains empty, then we can have (1, 1, 3) or (1,2,2) distribution pattern.
When boxes are kept in a row, they will be treated as different. In this case, the number of ways will be $^{5 - 1} C_{3 - 1} =^{4} C_{2} = 6$ .

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