In how many ways can Nicole distribute 20 chocolates among B

The source is "Ultimate guide to GMAT math" by Jay Stewart

I got the same answer in the book, what interests me is the logic behind this formula. thx!

Hello Bunuel,

Please why 20c4 does not work here? aren't we sure that at least each one will get a chocolate? can you explain the answer with a different method because I can't retain formulas. Thank you

 

­In how many ways can Nicole distribute 20 chocolates among Ben, Kate, Mark and Tom so that each receives at least one?

A. 20C4
B. 19C3
C. 21C4
D. 21C3
E. None of the answers is correct

When distributing identical objects, we should use the "Stars and Bars" method. 

"Stars and Bars" method:

Since each Ben, Kate, Mark and Tom must get at least 1 chocolate, we are essentially distributing only the remaining 20 - 4 = 16 chocolates.

Imagine the 16 chocolates as 16 stars in a row. To divide these stars among 4 people, we need 3 bars to create 4 separate sections. For example:

******|****|***|*** would represent that the first one got 6 chocolates, the second got 4, the third got 3, and the fourth got 3.

||**********|****** would represent that the first one got 0 chocolates, the second got 0, the third got 10, and the fourth got 6.

*******||******|*** would represent that the first one got 7 chocolates, the second got 0, the third got 6, and the fourth got 3.

So, the problem becomes one of arranging these 16 identical stars and 3 identical bars in a row, which is given by 19C3 = 19!/(16!3!) = 969.

Answer: B

Hope it helps.­

P.S. Check other questions related to distribuition in DISTRIBUTING ITEMS/PEOPLE/NUMBERS... (QUESTION COLLECTION) from our Special Questions' Directory.­