\( {}_{7}P_{x+36y} \)

The notation \( {}_{n}P_{r} \) typically represents the number of permutations of \( r \) items selected from \( n \) items, which is calculated using the formula: \[ {}_{n}P_{r} = \frac{n!}{(n-r)!} \] In your expression \( {}_{7}P_{x+36y} \), it seems you are looking for the number of permutations of \( x + 36y \) items selected from 7 items. However, for the permutation formula to be valid, \( r \) (in this case, \( x + 36y \)) must be less than or equal to \( n \) (which is 7). Thus, the expression \( {}_{7}P_{x+36y} \) is defined only if: \[ x + 36y \leq 7 \] If \( x + 36y \) exceeds 7, then \( {}_{7}P_{x+36y} \) is undefined (or can be considered as 0, since you cannot select more items than are available). If \( x + 36y \) is within the valid range, you can compute it using the formula: \[ {}_{7}P_{x+36y} = \frac{7!}{(7 - (x + 36y))!} \] If you have specific values for \( x \) and \( y \), you can substitute them into the expression to find the result.