C(n,m) and C(n+2,m+2)

 

\dbinom{ \,3 \,}{ \,2 \,} \; + \; 2 \, \dbinom{ \,2 \,}{ \,2 \,} \; = \; 5 \; = \; \dbinom{ \,5 \,}{ \,4 \,} \; = \; \dbinom{ \,3+2 \,}{ \,2+2 \,}

 
\dbinom{ \:6 \:}{ \:3 \:} \; +  \;2 \, \dbinom{ \:5 \:}{ \:3 \:} \; + \; 3 \, \dbinom{ \:4 \:}{ \:3 \:} \; + \; 4 \, \dbinom{ \:3 \:}{ \:3 \:} \; = \; 56 \; = \; \dbinom{ \:8 \:}{ \:5 \:} \; = \; \dbinom{ \:6+2 \:}{ \:3+2 \:}

 
 

Prove that

\dbinom{ \:n \:}{ \:m \:} \; + \; 2 \, \dbinom{ \:n-1 \:}{ \:m \:} \; + \; 3 \, \dbinom{ \:n-2 \:}{ \:m \:} \;+ \; ... \; + \; (n + 1 - m) \, \dbinom{ \:m \:}{ \:m \:} = \dbinom{ \:n+2 \:}{ \:m+2 \:}

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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