Graduate Management Admission Test (GMAT) Practice Test

When calculating arrangements of elements that include indistinguishable items, what is the formula used?

1. (Total arrangements) / (Number of indistinguishable elements)

2. (total number of arrangements) / (Factorial of number of indistinguishable elements)

3. (Distinct arrangements) / (Factorial of distinguishable elements)

4. (n! + k!) / (n - k)!

The correct answer is: (total number of arrangements) / (Factorial of number of indistinguishable elements)

The formula for calculating arrangements of elements that include indistinguishable items is based on the principle of permutations. When you have a total of n items, and some of those items are indistinguishable (meaning they are identical and cannot be differentiated from one another), the arrangements are calculated by taking the total number of arrangements (n!) and dividing that by the factorial of the number of indistinguishable elements. For example, if you have a set of letters such as A, A, B, you have a total of 3 letters, but since two of them are indistinguishable (the A's), the formula to find the distinct arrangements would be: Total arrangements = 3! (for three letters) Indistinguishable elements = 2! (for the two A's) Thus, the number of distinct arrangements becomes 3! / 2!. This reflects the need to account for the indistinguishability of certain items in the calculation. This reasoning leads to the conclusion that the correct formula is to take the total number of arrangements and divide by the factorial of the number of indistinguishable elements, which is precisely what is described in the correct choice.