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For m,n E Zt, define P(m, n) to be: (п + т - 1)! n!(m – 1)! the number of ways to write n = x1+ · ·.+ xm with x1, ... , xm E N is (a) . Prove each of the following statements. i. Vn e Zt, P(1, n). ii. Vm e Z+, P(m, 1). iii. Vm, n e Zt, P(m,n+ 1) ^ P(m +1, n) = P(m + 1, n + 1).

For m,n E Zt, define P(m, n) to be: (п + т - 1)! n!(m – 1)! the number of ways to write n = x1+ · ·.+ xm with x1, ... , xm E N is (a) . Prove each of the following statements. i. Vn e Zt, P(1, n). ii. Vm e Z+, P(m, 1). iii. Vm, n e Zt, P(m,n+ 1) ^ P(m +1, n) = P(m + 1, n + 1).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For m, n e Zt, define P(m, n) to be: (п + m — 1)! n!(m – 1)! the number of ways to write n = X1 + · .. + Xm with 1,..., xm E N is (a) Prove each of the following statements. i. Vn e zt, ii. Vm e Zt, P(m, 1). iii. Vm, n E Zt, P(m, n+ 1) ^ P(m + 1, n) → P(m +1, n + 1). P(1, п). (b) Use the results from part (a) to prove P(2,2) ^ P(3,3). (c) For t e Z+ witht > 2, define Q(t) to be: Vm,n E Z+,m +n = t → P(m, n). Prove, by Induction: Vt E Z+,t > 2 = Q(t). HINT: You may use the results from part (a) in your solution. (d) Use the results from previous parts in this question to prove Vm, n E Z*, P(m, n).
Transcribed Image Text:For m, n e Zt, define P(m, n) to be: (п + m — 1)! n!(m – 1)! the number of ways to write n = X1 + · .. + Xm with 1,..., xm E N is (a) Prove each of the following statements. i. Vn e zt, ii. Vm e Zt, P(m, 1). iii. Vm, n E Zt, P(m, n+ 1) ^ P(m + 1, n) → P(m +1, n + 1). P(1, п). (b) Use the results from part (a) to prove P(2,2) ^ P(3,3). (c) For t e Z+ witht > 2, define Q(t) to be: Vm,n E Z+,m +n = t → P(m, n). Prove, by Induction: Vt E Z+,t > 2 = Q(t). HINT: You may use the results from part (a) in your solution. (d) Use the results from previous parts in this question to prove Vm, n E Z*, P(m, n).
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