1. **Problem statement:** We want to find the number of nondecreasing sequences of natural numbers \(\langle a_1, a_2, \ldots, a_k \rangle\) with sum 49 that are *not* good. A sequence is good if there exists a permutation \(\langle b_1, b_2, \ldots, b_k \rangle\) of the same elements such that for every \(i\), the partial sum \(b_1 + \cdots + b_i \neq 24\) and \(\neq 25\). 2. **Understanding the problem:** - The sequence \(a_i\) is nondecreasing and consists of natural numbers. - The sum \(\sum a_i = 49\). - A sequence is *not* good if *every* permutation has a partial sum equal to 24 or 25 at some point. 3. **Key insight:** - If a sequence is not good, then no matter how we reorder it, the partial sums must hit 24 or 25. - Since the total sum is 49, the partial sums go from 0 up to 49. - The forbidden partial sums are exactly 24 and 25. 4. **Reformulating:** - The sequence is not good if for *every* permutation \(b\), there exists \(i\) such that \(\sum_{j=1}^i b_j = 24\) or 25. - Equivalently, the set of partial sums of any permutation must include 24 or 25. 5. **Considering the partial sums:** - Partial sums are sums of subsets of the multiset \(\{a_1, a_2, \ldots, a_k\}\) arranged in some order. - Since permutations reorder the sequence, the partial sums correspond to sums of prefixes. - The question reduces to: Is there a permutation whose prefix sums avoid 24 and 25? 6. **When is a sequence not good?** - If for *every* ordering, the partial sums hit 24 or 25, then the sequence is not good. - This means 24 or 25 is an unavoidable partial sum in any permutation. 7. **Key combinatorial fact:** - If the multiset can be partitioned into two parts with sums 24 and 25, then by ordering the 24-sum part first, the partial sum 24 appears. - Similarly, if any subset sums to 24 or 25, then by placing that subset first, the partial sum hits 24 or 25. 8. **Therefore:** - If the multiset has a subset summing to 24 or 25, then the sequence is not good. - Conversely, if no subset sums to 24 or 25, then there exists a permutation avoiding these partial sums, so the sequence is good. 9. **Counting sequences not good:** - We want the number of nondecreasing sequences of natural numbers summing to 49 that have at least one subset summing to 24 or 25. 10. **Summary:** - The problem reduces to counting nondecreasing sequences of natural numbers summing to 49 that contain a subset summing to 24 or 25. **Final answer:** The number of nondecreasing sequences of natural numbers summing to 49 that are not good equals the number of such sequences containing a subset summing to 24 or 25. (This problem is combinatorially complex and typically requires advanced subset-sum and partition counting techniques or computational enumeration.)