3. Recall that a composition of a positive integer n is a way of writing n as a sum of positive integers, called parts, which may appear in any order. It turns out to be interesting to count the number of compositions of n using only odd parts. Here is a table for small values of n: n 1 2 3 4 5 compositions of n with only odd parts 1 1+1 3, 1+1+1 3+1, 1+3, 1+1+1+1 5, 3+1+1, 1+3+1, 1+1+3, 1+1+1+1+1 In class we showed that each composition of n comes from a composition of n - 1 by doing one of two things: 2 1. adding 1 as a new last part 2. adding 1 to the current last part The first operation still gets us from a composition of n-1 with all parts odd to a composition of n with all parts odd. The second operation fails, but there is a replacement: add 2 to the current last part of a composition of n - 2 with all parts odd. Thus, for example, the compositions of 5 above with last part 1 come from adding 1 as a new last part to the compositions of 4 above. The compositions of 5 above whose last part is not 1 come from adding 2 to the last part of the compositions of 3 above. Recall that the Fibonacci numbers are defined by Fn+1 = Fn + Fn-1 for n ≥ 1, with Fo = 0 and F₁ = 1. Prove by induction that the number of compositions of n with all parts odd is Fn.

How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?