If `.^(n+1)C_(r+1):^(n)C_(r):^(n-1)C_(r-1)=11:6:3`, find the values of n and r.

Explanation

Here, `(.^(n+1)C_(r+1))/(.^(n)C_(r))=(11)/(6)`
`implies(n+1)/(r+1)*(.^(n)C_(r))/(.^(n)C_(r))=(11)/(6)" "[because.^(n)C_(r)=(n)/(r)*^(n-1)C_(r-1)]`
`implies(n+1)/(r+1)=(11)/(6)`
`implies6n+6=11r+11`
`implies6n-11r=5` . . . (i)
and `(.^(n)C_(r))/(.^(n-1)C_(r-1))=(6)/(3)`
`implies(n)/(r)*(.^(n-1)C_(r-1))/(.^(n-1)C_(r-1))=(6)/(3)" "[because.^(n)C_(r)=(n)/(r)*^(n-1)C_(r-1)]`
`implies(n)/(r)=2`
`impliesn=2r` . . (ii)
On solving Eqs. (i) and (ii), we get n=10 and r=5.