For $$non-negative$$ integers s and r, let $$sr=s$$!r!(s$$−$$r)! if r≤$$s0$$ if rs For positive integers m and n , Let $$g(m$$,$$n)=$$∑$$p=0m+n$$ $$f(m$$,n,$$p)n+ppwhere$$ for any nonnegative integer p,$$f(m$$,n,$$p)=$$∑$$i=0p$$ $$min+ipp+np$$−i Then which of the following statements $$is/are$$ TRUE?

Question

For $$non-negative$$ integers s and r, let $$sr=s$$!r!(s$$−$$r)! if r≤$$s0$$ if r>s For positive integers m and n ,  Let $$g(m$$,$$n)=$$∑$$p=0m+n$$ $$f(m$$,n,$$p)n+ppwhere$$ for any nonnegative integer p,$$f(m$$,n,$$p)=$$∑$$i=0p$$ $$min+ipp+np$$−i Then which of the following statements $$is/are$$ TRUE?

Infinity Learn · Accepted Answer

$$solvingf(m$$,n,$$p)=$$∑$$i=0p$$ $$mCin+iCp$$⋅$$p+nCp-inow$$ , mCi⋅$$n+iCp$$⋅$$p+nCp$$−$$i=$$ mCi⋅(n+i)!p!(n$$−$$p+i)!×(n+p)!(p$$−$$i)!(n+i)!$$=$$ mCi×(n+p)!p!×$$1(n$$−$$p+i)$$!(p$$−$$i)!$$=$$ mCi×(n+p)!p!n!×n!(n$$−$$p+i)!(p$$−$$i)!$$=$$ $$mCin+pCp$$⋅nCp−i mCi⋅nCp−$$i=m+nCpf(m$$,n,$$p)=n+pCp$$⋅$$m+nCpf(m$$,n,$$p)$$ $$n+pCp=m+nCp$$ Now $$g(m$$,$$n)=$$∑$$p=0m+n$$ $$f(m$$,n,$$p)$$ $$n+pCpg(m$$,$$n)=$$∑$$p=0m+n$$ $$m+nCpg(m$$,$$n)=2m+n$$ (A) $$g(m$$,$$n)=g(n$$,$$m)$$ (B) $$g(m$$,$$n+1)=2m+n+1g(m+1$$,$$n)=2m+1+n$$ (D) $$g(2m$$,$$2n)=22m+2n=2m+n2=(g(m$$,$$n))2$$

For non-negative integers s and r, let sr=s!r!(s−r)! if r≤s0 if r>s For positive integers m and n , Let g(m,n)=∑p=0m+n f(m,n,p)n+ppwhere for any nonnegative integer p,f(m,n,p)=∑i=0p min+ipp+np−i Then which of the following statements is/are TRUE?

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