Find the value (s) of `r` satisfying the equation `"^69 C_(3r-1)-^(69)C_(r^2)=^(69)C_(r^2-1)-^(69)C_(3r)dot`

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

.^((n-1)C_(r) + ^((n-1)) C_((r-1)) is

If n and r are two positive integers such that nger,"then "^(n)C_(r-1)+""^(n)C_(r)=

The roots of the equation |^x C_r^(n-1)C_r^(n-1)C_(r-1)^(x+1)C_r^n C_r^n C_(r-1)^(x+2)C_r^(n+1)C_r^(n+1)C_(r-1)|=0 are x=n b. x=n+1 c. x=n-1 d. x=n-2

Prove that ""^(n)C_r + ""^(n)C_(r-1) = ""^(n+1)C_r

If .^(15)C_(2r-1)=.^(15)C_(2r+4) find r

the roots of the equations |{:(.^(x)C_(r),,.^(n-1)C_(r),,.^(n-1)C_(r-1)),(.^(x+1)C_(r),,.^(n)C_(r),,.^(n)C_(r-1)),(.^(x+2)C_(r),,.^(n+1)C_(r),,.^(n+1)C_(r-1)):}|=0

If ""^(100)C_(r)=""^(100)C_(3r) then r is:

If ""^(n)C_(r) denotes the number of combinations of n things taken r at a time, then the expression ""^(n)C_(r+1)+""^(n)C_(r-1)+2xx""^(n)C_(r) , equals

Prove that if 1 le r le n " then " n xx^((n-1))C_(r-1)= (n-r+1).^(n)C_(r-1)