Counting principles

Questions 1 to 5 of 8

1. 1. How many non-negative integer solutions does x1 + x2 + x3 + x4 + x5 = 30 have, with no upper limits on any variable?

   Type a number.

2. 2. How many positive integer solutions does x1 + x2 + x3 + x4 = 9 have, where every variable is at least 1?

   Type a number.

3. 3. In inclusion-exclusion over the boxes that break their upper bound, what sign does the term for a set S of boxes get?

   Always +1, because each term counts solutions(-1) raised to the number of boxes in S, so the sign depends on whether that number is odd(-1) raised to k, the total number of boxes, for every S-1 whenever the remaining total R_S is negative
4. 4. There are k = 3 boxes. In the mask, bit i set means box i is in S. Which masks get a minus sign in the sum?

   Pick every answer that is true.

   0b0010b0110b1010b111
5. 5. A student changes one line of count_bounded in the bakery solution. The sample input 30 5 with the five limits is run again. ``cpp ans = (__builtin_popcount(mask) % 2 == 0) ? (ans - term + MOD) % MOD : (ans + term) % MOD; `` What does the changed program print, modulo 1000000007?

   2376999997631463760