Question

C & J Realty has received 12 inquiries from prospective home buyers. In how many ways can the inquiries be directed to any three of the firm's real estate agents if each agent handles four inquiries? ——- ways

Answer 1

There are 55 ways to distribute 12 inquiries among three real estate agents, with each agent handling four inquiries. This is determined using combinatorics by treating the problem as choosing positions for dividers in a sequence of inquiries to partition them into three equal parts.

Step-by-step explanation:

The question asks us to determine the number of ways to distribute 12 inquiries among three real estate agents, with each agent handling exactly four inquiries. This is a problem of distributing distinct objects into identical groups without regard to the order within each group. Since each agent is to get the same number of inquiries, this is a partitioning problem which can be solved using combinatorics.

To solve this, we treat each inquiry as a distinct object and each agent as an identical box. We are essentially arranging the inquiries in a row, and then choosing points to divide the row into three equal parts, each part corresponding to one agent. The dividers can be placed in any of the slots between the inquiries. That means we have 11 slots where we can place two dividers, because we have 12 � 1 = 11 slots between the inquiries.

So, the number of ways to divide the 12 inquiries among the three agents is the same as the number of ways to choose 2 positions for the dividers out of the 11 possible positions, which is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the number of slots and k is the number of dividers. In this case:

C(11, 2) = 11! / (2!(11-2)!) = 55

Therefore, there are 55 ways the inquiries can be directed to any three of the firm's real estate agents if each agent handles four inquiries.

Answer 2

The number of ways the inquiries directed = 34,650

Step-by-step explanation:

Here we have to use the permutation.

If number of different permutations of "n" objects where there are
`n_1` repeated items,
`n_2` ....
`n_k` repeated items, then the number of ways =
`(n!)/(n_1! n_2!...n_k!)`

Given:

Then number calls received = 12. So n = 12

The number of real estate agents = 3 and each agent handles 4 inquiries.

The number of ways the inquiries directed =
`(12!)/(4!.4!.4!)`

=
`(479,001,600)/(24.24.24)`

=
`(479,001,600)/(13824)`

= 34,650 ways

So, the number of ways the inquiries directed = 34,650