Question 16

# Given below are two statements :Statement I: The number of ways to pack six copies of the same book into four identical boxes where a box can contain as many as six books, is 9. Statement II: The minimum number of students needed in a class to guarantee that there are at least six students whose birthday fall in the same month, is 61 . In the light of the above statements, choose the correct answer from the options given below.

Solution

Statement 1

We need to pack 6 indistinguishable books into 4 indistinguishable boxes

The ways we can pack the books are

6 ,0,0,0 (only one way as 0,6,0,0 or 0,0,6,0 won’t be different as boxes are identical)
5, 1,0,0
4, 2,0,0
4, 1, 1,0
3, 3,0,0
3, 2, 1,0
3, 1, 1, 1
2, 2, 2,0
2, 2, 1, 1.

so total 9 ways

Hence statement 1 is correct.

Statement 2

To calculate the minimum number of students required, we need to consider the worst case scenario i.e. the birthdays of students are evenly spread across all the 12 months. We'll have to assume that there are atleast 5 birthday is each month, so the total number of students required = 5* 12 = 60. To assure that there is 1 month in which we have 6 birthdays, we'll have to add 1 to this number, hence the minimum number of students required = 60 + 1 = 61

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