MathLabs

Problem 2

Let m,nm,n be positive integers with n≤mn\le m. Prove that 2nn!≤(m+n)!(m−n)!≤(m2+m)n2^n n!\le\dfrac{(m+n)!}{(m-n)!}\le(m^2+m)^n.
Step 2 of 4: Simplify a paired factor
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(m+i)(m−i+1)=m2+m−i2+i(m+i)(m-i+1)=m^2+m-i^2+i
Detailed analysis

Direct expansion gives (m+i)(m−i+1)=m2+m−i2+i(m+i)(m-i+1)=m^2+m-i^2+i. This is the quantity to estimate for each ii from 11 to nn.