Question

Five friends have a total of 12 identical pencils, and each one has at least two pencils. In how many ways can this happen?

(A) 10
(B) 13
(C) 16
(D) 15
(E) 12

Answer 1

The total number of ways to distribute 12 identical pencils among five friends, given that each has at least two pencils, is 9 (by giving 2 pencils to each initially and distributing the remaining 2). The answer is not present in the listed options.

Step-by-step explanation:

The question asks for the number of ways five friends can share 12 identical pencils, given that each friend has at least two pencils. This problem is a combinatorics question that can be solved by using partitions of an integer. Since each person must have at least two pencils, we initially give each person two pencils, hence using 10 pencils in total. Now we need to distribute the remaining 2 pencils among the 5 friends.

We can do this distribution in 6 different ways:

• 2 people get 1 extra pencil each (4 different scenarios)
• 1 person gets both pencils (5 different scenarios)

The total number of ways is therefore 4 + 5 = 9, which is not listed in the given options (A) to (E).