If `\ ^(43)C_(r-6)=\ ^(43)C_(3r+1)\ ` then the value of `r` is

Explanation

We have, `((18),(r-2))+2((18),(r-1))+((18),(r))ge((20),(13))`
It means that `.^(18)C_(r-2)+2*.^(18)C_(r-1)+.^(18)C_(r)ge.^(20)C_(13)`
`implies(.^(18)C_(r-2)+.^(18)C_(r-1))+(.^(18)C_(r-1)+.^(18)C_(r))ge.^(20)C_(7)`
`implies.^(19)C_(r-1)+.^(19)C_(r) ge .^(20)C_(7)`
`implies.^(20)C_(r)ge.^(20)C_(7)`
`implies.^(20)C_(r) ge .^(20)C_(13)`
`implies 7 le r le 13`
`therefore r=7,8,9,10,11,12,13`
Hence, the number of values of r are 7.