If `""^(13)C_(X) = ""^(13)C_(Y)` and X `ne` Y , then the value of X+Y is

Options

A. 12
B. 13
C. 26
D. 7

Answer

To solve the problem, we start with the given equation:

\[ ^{13}C_{X} = ^{13}C_{Y} \]

Since \(X \neq Y\), we can use the property of combinations that states:

\[ ^{n}C_{r} = ^{n}C_{n - r} \]

Applying this property to our equation, we can rewrite \(^{13}C_{X}\) as:

\[ ^{13}C_{X} = ^{13}C_{13 - X} \]

Now, we can set the two expressions equal to each other:

\[ ^{13}C_{13 - X} = ^{13}C_{Y} \]

Since both sides are equal, we can conclude that:

\[ 13 - X = Y \]

Now, we want to find \(X + Y\). We can substitute \(Y\) from the equation we derived:

\[ X + Y = X + (13 - X) \]

Simplifying this gives:

\[ X + Y = 13 \]

Thus, the value of \(X + Y\) is:

\[ \boxed{13} \]