Consider all subsets of ${1,2,\dots,A+1}$ of cardinality $A-s+1$. There are exactly $\binom{A+1}{s}$ wayssubsets. For each such subset $x_1 < x_2 < \dots < x_{A-s+1}$ consider the element $x_{A-n+1}$. The number of subsets with fixed value $x_{A-n+1}=p:=A-n+j+1$ the number of desired subsets equals $\binom{A-n+j}{A-n}\binom{n-j}{n-s}=\binom{A-n+j}{j}\binom{n-j}{i}$. Then just sum up by all possible values of $j=x_{A-n+1}-A+n-1$.
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Consider all subsets of ${1,2,\dots,A+1}$ of cardinality $A-s+1$. There are exactly $\binom{A+1}{s}$ ways. For each such subset $x_1 < x_2 < \dots < x_{A-s+1}$ consider the element $x_{A-n+1}$. The number of subsets with fixed value $x_{A-n+1}=p:=A-n+j+1$ the number of desired subsets equals $\binom{A-n+j}{A-n}\binom{n-j}{n-s}=\binom{A-n+j}{j}\binom{n-j}{i}$. Then just sum up by all possible values of $j=x_{A-n+1}-A+n-1$. |
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