Question

m+nP2, = 56, m-nP2, = 30 then
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Answer 1

Given:

ᵐ⁺ⁿP₂ = 56

ᵐ⁻ⁿP₂ = 30

To find:

(m,n)

Answer:

We know, ⁿPᵣ = $\frac{(n)!}{(n-r)!}$

∴ ᵐ⁺ⁿP₂ = $\frac{(m+n)!}{(m+n-2)!}$     ----------(1)

and

ᵐ⁻ⁿP₂ = $\frac{(m-n)!}{(m-n-2)!}$       ----------(2)

(m+n)! = (m+n)(m+n-1)(m+n-2)(m+n-3)...   ----------(3)

(m-n)! = (m-n)(m-n-1)(m-n-2)(m-n-3)...       ----------(4)

(m+n-2)! = (m+n-2)(m+n-3)(m+n-4)...        ----------(5)

(m-n-2)! = (m-n-2)(m-n-3)(m-n-4)...           ----------(6)

Substitute (3) and (5) in (1) and (4) and (6) in (2)

We get:

ᵐ⁺ⁿP₂ =  $\frac{(m+n)(m+n-1)(m+n-2)(m+n-3)...}{(m+n-2)(m+n-3)(m+n-4)...}$ = 56

ᵐ⁻ⁿP₂ = $\frac{ (m-n)(m-n-1)(m-n-2)(m-n-3)... }{(m-n-2)(m-n-3)(m-n-4)... }$ = 30

Dividing, we get:

ᵐ⁺ⁿP₂ = (m+n)(m+n-1) ----------(7)

ᵐ⁻ⁿP₂ = (m-n)(m-n-1)  -----------(8)

Trying to solve this will result in a Quadratic equation in two variables/ To avoid that, consider:

m+n = u ----------(9)

m-n = v  ----------(10)

Substitute (9) in (7) and (10) in (8)

ᵐ⁺ⁿP₂ = u(u-1)

ᵐ⁻ⁿP₂ = v(v-1)

ᵐ⁺ⁿP₂ = u² - u = 56

u² - u = 56

u² - u - 56 = 0

Splitting the middle term

u² - 8u + 7u - 56 = 0

u(u - 8) + 7(u - 8) = 0

u + 7 = 0

u - 8 = 0

∴ u = -7 and 8

But, u ≠ -7 ∵ u = n ∈ N

∴u = 8

Similarly, m-n = v

∴ v(v-1) = 30

v² - v = 30

v² - v - 30 = 0

Splitting the middle term,

v² - 6v + 5v - 30 = 0

v(v - 6) + 5(v - 6) = 0

v + 5 = 0

v - 6 = 0

∴ v = -5 and 6

But v ≠ -5 ∵ v = n ∈ N

∴ v = 6 = m-n

∴ m + n = u = 8 ----------(11)

  m - n = v = 6  ----------(12)

Add (11) and (12)

m + n + m - n = 8 + 6

2m = 14

m = 7

∵ m + n = 8,

∴ 7 + n = 8

n = 1

∴(m,n) = (7,1)

Hope it helps, byeeee