chalte chalte combinatorics 6

^k-1C_k-1 + ^kC_k-1 +........+^n+k-2C_k-1=?

Answer-^{n+ k-1}C_k

2 Answers

^k-1C_k-1= ^kC_k

now ^kC_k+ ^kC_k-1=^k+1C_k

our series is
^k-1C_k-1+^kC_k-1+^k+1C_k-1+...+^n+k-2C_k-1
= ^k+1C_k +^k+1C_k-1+...+^n+k-2C_k-1
=^k+2C_k+...+^n+k-2C_k-1
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=^n+k-1C_k

thanks!!`

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