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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`

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The number of values of 'r' satisfying the equation ""^(39)C_(3r-1)- ""^(39)C_(r^(2) )= ""^(39)C_(r^(2)-1) - ""^(39)C_(3r) is

.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)=

""^(n)C_(r+1)+^(n)C_(r-1)+2.""^(n)C_(r)=

""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=

The number of values of r satisfying the equation 69 C 3r−1 ​ − 69 C r 2 ​ = 69 C r 2 −1 ​ − 69 C 3r ​ is:

""^(n)C_(r)+2""^(n)C_(r-1)+^(n)C_(r-2) is equal to

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

If nger , then ""^(r)C_(r)+""^(r+1)C_(r)+""^(r+2)C_(r)+......+""^(n)C_(r) is equal to

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

If .^(15)C_(3r)=.^(15)C_(r+3), find .^(r)C_(2) .

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