# Permutations & Combinations - Questions & DiscussionsQuantitative Report

.Can someone help me out with the following problems with some clear and simple approcah to these qustions
4 balls are to be put in 5 boxes.in how many ways can this be done if
a)Balls are similar and boxes are different
b)balls are different and boxes are similar
c)both boxes and ...
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Alka.Maheshwari (n+r-1)Cr-1 will be applied when you have to find non neg.... 07 Jun.
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Page 44 of 45
Page 45 of 45
4 balls are to be put in 5 boxes.in how many ways can this be done if
a)Balls are similar and boxes are different
This is similar to finding the whole number solutions for the equation
a + b + c + d + e = 4
which is given by (n+r-1) C r-1 where n is the constant on the RHS and r is the number of variables.
Hence in this case we have 8 C 4 = 70 ways

c)both boxes and balls are similar
Here its the number of ways you can split 4 into 5 numbers
40000
31000
22000
21100
11110
Hence 5 ways

b)balls are different and boxes are similar
For each of the cases listed in (c) the number of ways of selecting the balls has to be considered
For eg, for the case 40000, the 4 balls for the first box can be selected in only 1 way.
For the case 31000 the 3 balls for the first box can be selected in 4C3 = 4 ways
In 22000 2 balls for the first box can be selected in 4C2 = 6 ways (note the remaining 2 balls automaticallly select themselves for the second box). Similarly in 21100 the 2 balls to go together can be selected in 6 ways, the remaining two balls automatically go into different boxes.Finally, in 11110 there is just 1 way of splitting all the balls into 4 boxes. Hence, total number of ways = 1+4+6+6+1 = 18 ways

Note: The number of positive integral solution(natural number) for
x1+x2+x3+.........xr=n is (n-1) C (r-1)