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Find the number of monomials of total degree exactly 12 in five…
Question
Find the number of monomials of total degree exactly 12 in five variables x1, x2, x3, x4, x5 in which every individual exponent is at most 6. Equivalently, count the nonnegative integer solutions of e1+e2+e3+e4+e5 = 12 with each ei <= 6.
✓ Verified answer: 1190checked by our engine — not a guess
Step-by-step solution
Unrestricted nonnegative solutions of e1+...+e5 = 12 number C(16,4) = 1820.
Apply inclusion-exclusion to enforce ei <= 6, i.e.
exclude ei >= 7.
Forcing one variable >= 7 (substitute ei' = ei-7) leaves sum 5 among 5 variables: C(9,4) = 126, with C(5,1) = 5 choices, giving 5*126 = 630.
Forcing two variables each >= 7 removes 14 > 12: impossible, contributes 0.
Thus the count = 1820 - 630 = 1190.
Final answer1190
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