Question

In how many ways can one a, three b's, two c's, and one f be distributed among seven students in a class?

Answer 1

There are 420 different ways to distribute one 'a', three 'b's, two 'c's, and one 'f' among seven students, calculated using permutations with indistinguishable objects.

Step-by-step explanation:

The question is asking in how many different ways we can distribute one ‘a’, three ‘b’s, two ‘c’s, and one ‘f’ among seven students. This problem can be approached using the concept of permutations where repetition is allowed for some items. Using the formula for permutations with indistinguishable objects, we calculate the total number of distributions as:

Here, 7! is the factorial of 7 which calculates the total number of permutations without considering the redundancy caused by the identical items. 3! accounts for the permutations of three indistinguishable ‘b’s, and 2! accounts for the two indistinguishable ‘c’s. We perform the calculation:

(7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 )

(3! = 3 × 2 × 1 = 6 )

(2! = 2 × 1 = 2 )

The division of the total permutations (5040) by the permutations of the ‘b’s (6) and the ‘c’s (2) gives us the result:

Thus, there are 420 different ways to distribute the letters among the students.