+306 (a) Count the number of quadruples (a, b, c, d) of nonnegative integers such that 0≤a<b<c<d≤12.
(b) For this part, we want to count the number of quadruples (a, b, c, d) of nonnegative integers such that 0≤a≤b≤c≤d≤12.
Here, some of a, b, c, and d can be equal to each other, so the answer will be different from part (a). Each value a, b, c, d must be between 0 and 12 inclusive. One idea is to count how many times each number appears.
For example, suppose (a, b, c, d)=(1, 3, 3, 8). Then we can make a table that counts how many times each number appears among a, b, c, and d: 01234567891011120101000020000
Use this idea to find the number of quadruples (a, b, c, d). (If you come up with a different approach, you are free to use it.)
(c) In general, find the number of k-tuples (a1,a2,a3,…,ak) of nonnegative integers such that 0≤a1≤a2≤a3≤⋯≤ak≤n.
https://web2.0calc.com/questions/a-count-the-number-of-quadruples-a-b-c-d-of-nonnegative
+6251 Any set of 4 distinct integers can be put in strictly ascending orderWe can choose (134)=715 sets of distinct integers in [0,12]
b) is the same idea as (a) but now the integers don't have to be distinctwe have a few flavors of how the selected integers might appeari) 4 distinct integers, ii) 1 pair, ii) 2 pairs, iv) 3 of a kind, v) 4 of a kindFor all these flavors there is only a single way they can be distinctly sorted in ascending orderso N=(134)+13(122)+(132)+13⋅12+13=1820where each of those terms corresponds to the number of each flavor that occurs
you can do (c)
