Question

Q6
Consider three boxes, each containing 10 balls labelled
1,2.., 10. Suppose one ball is randomly drawn from each of
the boxes. Denote by ni, the label of the ball drawn from the
i th box, (i=1,2,3). Then, the number of ways in which the
balls can be chosen such that n1 <n2 <n3 is:
(1) 120
(2) 82
(3) 240
(4) 164​

Answer 1

Given :  three boxes, each containing 10 balls labelled  1,2.., 10.

To find : the number of ways in which the  balls can be chosen such that n1 <n2 <n3 is:

Solution:

Box 1      Box  2       Box

1              2              3 - 10     ( 8 ways)

               3             4 - 10      ( 7 ways)

---.......

               9             10    ( 1 way)

Number of ways = 8 + 7 + ---------+ 2 + 1

= 36

Box 1      Box  2       Box

2              3              4 - 10     (7 ways)

               4              5 - 10      ( 6 ways)

---.......

               9             10    ( 1 way)

Number of ways = 7 + ---------+ 2 + 1

= 28

Box 1      Box  2       Box

3             4             5 - 10     (6 ways)

               5              6 - 10      ( 5 ways)

---.......

               9             10    ( 1 way)

Number of ways = 6 + ---------+ 2 + 1

= 21

and so  on

Total ways =

36  + 28  + 21  + 15 +  10  +  6  + 3  + 1

= 120

120 Ways  in which the  balls can be chosen such that n1 <n2 <n3  

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