Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the
three people has at least two apples?
(A) 105 (B) 114 (C) 190 (D) 210 (E) 380

Question

Accepted Answer

190 (C). Step 1: Subtract the minimum apples each person 
must have, which is 2 apples each. So, 24-2*
3=18 apples remain to be distributed. 
Step 2: Use the stars and bars method to distribute 
the remaining apples. There are two partitions 
(between Alice, Becky, and Chris) and 18 identical 
apples. 
Step 3: The formula for stars and bars is beginpmatrix n+k-1 k-1endpmatrix , 
where n is the number of items to distribute and k is 
the number of partitions. Here, n=18 and k=3. 
Step 4: Calculate the combinations: beginpmatrix 18+3-1 3-1endpmatrix =
beginpmatrix 20 2endpmatrix. 
Step 5: Calculate beginpmatrix 20 2endpmatrix which is  (20* 19)/2* 1 =190. 
The answer is 190, which corresponds to option (C).

Question

Question

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? (A) 105 (B) 114 (C) 190 (D) 210 (E) 380

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