(A) We know that if $^{n}C_{x} = ^{n}C_{y}$,then either $x = y$ or $x + y = n$.Given $^{15}C_{3r} = ^{15}C_{r+3}$.Case $1$: $3r = r + 3$ $\Rightarrow

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(A) We know that if $^{n}C_{x} = ^{n}C_{y}$,then either $x = y$ or $x + y = n$.Given $^{15}C_{3r} = ^{15}C_{r+3}$.Case $1$: $3r = r + 3$ $\Rightarrow 2r = 3$ $\Rightarrow r = 1.5$.Since $r$ must be an integer,this case is not possible.Case $2$: $3r + (r + 3) = 15$ $\Rightarrow 4r + 3 = 15$ $\Rightarrow 4r = 12$ $\Rightarrow r = 3$.Thus,the value of $r$ is $3$.