1.

If `""^(n)C_(r -1) = 36, ""^(n)C_(r) = 84 " and " ""^(n)C_(r +1) = 126`, then find the value of `""^(r)C_(2)`.

Answer» Correct Answer - n = 9 and r = 3
We know that, `(""^(n)C_(r))/(""^(n)C_(r - 1)) = (n - r + 1)/(r)`
`rArr " " 84/36 = 7/3 = (n - r + 1)/(r) " "` [given]
`rArr " " 3n - 10r + 3 = 0 " "….(i)`
Also given, `(""^(n)C_(r))/(""^(n)C_(r + 1)) = 84/126`
`rArr " " (r + 1)/(n - r) = 2/3`
`rArr" " 2n - 5r - 3 = 0 " "..(ii)`
On solving Eqa. (i) and (ii), we get
r = 3 and n = 9


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