A composition of the number n is an ordered set of integers a1,a2,…,ak such that a1+a2+⋯+ak=n. The numbers a1,a2,…,ak are called the parts of the composition. Determine the number of compositions of 23 where each part is at least 4.
To determine the number of compositions of 23 where each part is at least 4, we can use a technique called dynamic programming.
Let's define a function C(n, k) as the number of compositions of n with exactly k parts, where each part is at least 4. We can find the solution to our problem by evaluating C(23, k) for k ranging from 1 to 23 and then summing up the results.
To compute C(n, k), we can use the following recurrence relation:
C(n, k) = C(n-4, k-1) + C(n-5, k-1) + C(n-6, k-1) + ... + C(n-k, k-1)
This means that the number of compositions of n with exactly k parts, where each part is at least 4, can be obtained by summing up the number of compositions of n minus each possible part size (from 4 to k) with k-1 parts.
We can start by calculating C(n, 1), which represents the number of compositions of n with exactly 1 part. In this case, since there are no restrictions on the parts, C(n, 1) = 1 for all values of n.
Next, we can calculate C(n, 2) using the recurrence relation. For example, to find C(23, 2), we calculate:
C(23, 2) = C(19, 1) + C(18, 1) + C(17, 1) + ... + C(2, 1)
We continue this process for values of k from 3 to 23, using the previously computed values of C(n, k-1).
Finally, we sum up the values of C(23, k) for k ranging from 1 to 23 to obtain the total number of compositions of 23 where each part is at least 4.
Note: This approach assumes that the order of the parts in a composition matters (e.g. [4, 4, 15] is different from [15, 4, 4]). If the order does not matter, you would need to divide the final result by the number of possible orderings of k parts (which is k!).
I hope this explanation helps you understand how to determine the number of compositions of 23 where each part is at least 4 using dynamic programming.