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).

As per our guidelines, we are supposed to answer only one question(3 subparts) and rest can…

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).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.6: Matrices

Problem 6TFE

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Question

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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