No. of ways to choose the 3 months =
C3 = 220.
Now, we have 6 people and 3 months for the birthdays to fall in.
Since none of these three months can have no birthday in it, the 3 possible ways of distributing the 6 people in 3 months would be
(4, 1, 1), (3, 2, 1), (2, 2, 2) (4,1, 1)
No. of ways of rotating the distribution of number of birthdays in these three months =
No. of ways of rotating the distribution of the people =
C4.C1.C1.
Therefore, total number of ways in (4, 1, 1) case =
..C4.C1.C1 = 90 ways.
(3, 2, 1)
No. of ways of rotating the distribution of number of birthdays in these three months = 3!
No. of ways of rotating the distribution of the people =
C3.C2.C1,
Therefore, total number of ways in (3, 2, 1) case =
3!.C3.C2.C1 = 360 ways.
(2, 2, 2)
No. of ways of rotating the distribution of number of birthdays in these three months =
No. of ways of rotating the distribution of the people =
C2.C2.C2 Therefore, total number of ways in (2, 2, 2) case
=
C2.C2.C2 = 90 ways.
The number of ways so that the birthdays of 6 people falls in exactly 3
calendar months = (90+360+ 90)- 220 = 118800