the birthday problem with seasons.( doubt)

[Mauli]

q) for a group of 7 people , find the probability that all 4 seasons ( winter, spring, summer, fall) occur at least once each among their birthdays , assuming that all seasons are equally likely.

[Sinistral]

It can be solved using Multinomial distribution.
It was too cumbersome to write 'here 'all the possible combinations so I am attaching a JPG image. I hope the hand-writing is readable.

[Mauli]

god!
thankyou so so much for these inputs sinistral.
one thing, i  had found a question set over the net and this question was there under the head of ''bayes theorem''.
if anything strikes u on that front please share it.
thanks again:)