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 recursive approach known as dynamic programming.

Let's define a function f(k, n) that represents the number of compositions of n where each part is at least k. We are interested in finding f(4, 23), as we want each part to be at least 4.

To compute f(k, n), we can consider two cases:

1. If k is greater than n, then it is impossible to form a composition where each part is at least k. Hence, f(k, n) = 0.

2. If k is less than or equal to n, we have two sub-cases to consider:
a) If k is equal to n, there is only one composition possible, which is [n].
b) If k is less than n, we can break down the problem into two sub-problems:
i) Fix the first part of the composition as k, which means we have to find the compositions of (n-k) with each part at least k.
ii) Don't fix the first part as k, which means we have to find the compositions of n with each part at least k+1.

Mathematically, f(k, n) = 1 + f(k, n-k) + f(k+1, n).

Using this recursive formula, we can start from smaller values of k and n and build our way up to finding f(4, 23), which will give us the desired number of compositions.

Let's use a table approach for dynamic programming to avoid redundant calculations. We initialize a 2D table with dimensions (kMax+1) * (nMax+1), where kMax is the maximum value of k, and nMax is the maximum value of n we are interested in. We fill in the table bottom-up, calculating the values of f(k, n) iteratively.

Here's a Python code snippet that implements this approach:

kMax = 4
nMax = 23

# Initialize the table
dp = [[0] * (nMax + 1) for _ in range(kMax + 1)]

# Base case: f(k, n) = 0 if k > n
for k in range(1, kMax + 1):
for n in range(k, nMax + 1):
if k == n:
dp[k][n] = 1
else:
dp[k][n] = 1 + dp[k][n - k] + dp[k + 1][n]

# Get the result
result = dp[kMax][nMax]
print(result)

After running this code, you will get the number of compositions of 23 where each part is at least 4.