+275 In how many ways can a teacher distribute 13 stickers to 6 of their students, if the three youngest students are triplets and insist on receiving an equal number of stickers? (The teacher does not need to give a sticker to each student.)
The only way i can think of is finding all the combinations and then adding them.
+1632 If the three youngest students in the class are triplets, then I think the problem is implying we can assume they are identical. If that is so, we can group those 3 into "1 person" or 1 variable, named "a". The other 3 students will be "b, c, d". Now it is stars and bars and we can do how to distribute 13 stickers among 4 people. a + b + c +d = 13, and a, b, c, and d are nonnegative, we do (13 + 4 - 1 choose 4 - 1)
The answer is 16 choose 3 or 560.
:)
+2668 Here's my approach:
Each of the triplets receives 0 stickers: Then there are 13 stars and 2 bars so there \({15 \choose 2} = 105\) ways to distribute the stickers.
Each of the triplets receives 1 sticker: Then there are 10 stars and 2 bars for \({12 \choose 2} = 66\) ways to distribute the stickers.
Each of the triplets receives 2 stickers: Then there are 7 stars and 2 bars for \({9 \choose 2} = 36\) ways to distribute the stickers.
Each of the triplets receives 3 stickers: Then there are 4 stars and 2 bars for \({6 \choose 2} = 15\) ways to distribute the stickers.
Each of the triplets receives 4 stickers: Then there are 1 star and 2 bars for \({3 \choose 2} = 3\) ways to distribute the stickers.
So, there are \(105 + 66 + 36 + 15 + 3 = \color{brown}\boxed{225}\) ways to distribute the stickers.
