Find the number of solutions in integers to n 1 + n 2 + n 3 + n 4 = 12 satisfying 0 ≤ n 1 ≤ 4 , 0 ≤ n 2 ≤ 5 , 0 ≤ n 3 ≤ 8 , and 0 ≤ n 4 ≤ 9.

Name: Find the number of solutions in integers to n 1 + n 2 + n 3 + n 4 = 12 satisfying 0 ≤ n 1 ≤ 4 , 0 ≤ n 2 ≤ 5 , 0 ≤ n 3 ≤ 8 , and 0 ≤ n 4 ≤ 9.
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find the number of solutions in integers to n 1 + n 2 + n 3 + n 4 = 12 satisfying 0 ≤ n 1 ≤ 4 , 0 ≤ n 2 ≤ 5 , 0 ≤ n 3 ≤ 8 , and 0 ≤ n 4 ≤ 9.

Question

Find the number of solutions in integers to       n    1    +      n    2    +      n    3    +      n    4    =  12 satisfying   0  ≤      n    1    ≤  4  ,  0  ≤      n    2    ≤  5  ,  0  ≤      n    3    ≤  8  ,  and   0  ≤      n    4    ≤  9.

grubijanebb · Accepted Answer

Step 1Let A denote nonnegative solutions to       n    1    +      n    2    +      n    3    +      n    4    =  12 and       A    i   denote those solutions where the upper bound on       n    i   is violated, then inclusion/exclusion gives       |                      A        1            ∪              A        2            ∪              A        3            ∪              A        4              ¯        |    =      |    A      |    −  (      |        A    1        |    +      |        A    2        |    +      |        A    3        |    +      |        A    4        |    )  +  (      |        A    1    ∩      A    2        |    +  ⋯  )  −  ⋯    =                    (                            12        +        4        −        1            12                      )              −      (                            (                                      7          +          4          −          1                7                              )                      +                            (                                      6          +          4          −          1                6                              )                      +                            (                                      3          +          4          −          1                3                              )                      +                            (                                      2          +          4          −          1                2                              )                      )    +                    (                            1        +        4        −        1            1                      )             where e.g.       |        A    1        |    =                    (                            7        +        4        −        1            7                      )             Step 2Because the solutions there can be recast as solutions to   (      m    1    +  5  )  +      n    2    +      n    3    +      n    4    =  12 with       m    1    ,      n    i    ≥  0 or       m    1    +      n    2    +      n    3    +      n    4    =  7, and omitted terms are because they are empty (i.e. they reduce to a sum of nonnegative integers to a negative integer).