Question

Counting rearrangements of items requires special consideration when items are not all different. Explain why the seven letters a, b, c, d, e, f, g can be rearranged in 7! ways, while a, a, a, d, e, f, g can be arranged in 7!÷3! ways.

a) The repeated letters in the second set reduce the total arrangements.
b) The order of the repeated letters doesn't affect the arrangements.
c) Repeated letters in both sets create more possibilities.
d) The number of letters is the only factor in arrangement.

Answer 1

The seven unique letters can be arranged in "7!" ways, while the set with three repeated letters (a, a, a, d, e, f, g) can be arranged in fewer ways, "7! ÷ 3!", due to the repeated letters not contributing to distinct arrangements.

Step-by-step explanation:

When arranging items where all elements are unique, such as the letters a, b, c, d, e, f, g, we can use the factorial notation, represented by "7!", to calculate the number of possible rearrangements. To get this, we multiply the number of choices for the first position by the number of remaining choices for the second position, and so on, which leads to "7 × 6 × 5 × 4 × 3 × 2 × 1" or "7!".

However, when some items are identical, as with the letters a, a, a, d, e, f, g, the order of the repeated items doesn't create distinct outcomes. Therefore, we adjust the calculation by dividing out the permutations of the identical items ('3!" for the three a's) from the total permutations. This results in the calculation "7! ÷ 3!", which accurately accounts for the reduced number of unique arrangements due to the repetition of the letter 'a'.