Question

i have 16 pieces of halloween candy that i want to distribute among 5 of the neighborhood children such that no one goes home empty-handed, i.e., each of them gets at least one piece of candy. in how many ways can i distribute the candy?

Answer 1

There are 364 ways to distribute 16 pieces of Halloween candy among 5 neighborhood children with each getting at least one piece, which is calculated using the stars and bars combinatorial method.

Step-by-step explanation:

To figure out how many ways there are to distribute 16 pieces of Halloween candy among 5 neighborhood children with each receiving at least one piece, we can use combinatorics. Since we want to ensure that no child goes empty-handed, we give one piece of candy to each child first. That leaves us with 16 - 5 = 11 pieces of candy to distribute freely among the 5 children.

We can think of this problem as a variation of the stars and bars method where we need to place four bars to create five spaces, representing the five children, into which we can distribute the remaining 11 candies (the stars). The number of ways we can arrange the candy is equal to the number of ways we can position the 11 stars and 4 bars in a row, which is calculated using the combination formula C(n+r-1, r-1), where n is the number of stars and r is the number of bars.

The formula gives us C(11+4-1, 4-1) = C(14, 3). This equals to 14!/(3!11!) which simplifies to 364. Therefore, there are 364 ways to distribute the candy among the 5 children.

Answer 2

There are 364 ways to distribute 16 pieces of Halloween candy among 5 neighborhood children.

Step-by-step explanation:

To distribute the 16 pieces of Halloween candy among the 5 neighborhood children, we can use the concept of 'stars and bars' or stars and separations. The stars represent the pieces of candy, and the bars represent the separations between the children. Since each child must receive at least one piece of candy, we can distribute the remaining 11 pieces any way we want among the 4 separations between the children.

The number of ways to distribute the 11 remaining candy pieces among the 4 separations can be calculated using the formula (n + k - 1) choose (k - 1), where n is the number of candy pieces and k is the number of separations. In this case, n = 11 and k = 4, so the number of ways to distribute the candy is (11 + 4 - 1) choose (4 - 1), which simplifies to 14 choose 3.

Using the formula for combinations, we have (14!)/(3!(14-3)!) = (14!)/(3!11!) = (14 x 13 x 12)/(3 x 2 x 1) = 364 ways to distribute the candy among the 5 children.