Find the value of 7 P 6 + 7 P 5 / 7 P 6 + 7 P 5

Question

ScanSolve AI Tutor · Accepted Answer

To find the value of the expression, we first observe its structure. The numerator and the denominator are identical.

Recognize that the expression is of the form $\frac{X}{X}$ .
The given expression is $\frac{{_7P_6} + {_7P_5}}{{_7P_6} + {_7P_5}}$ .
Let $X = {_7P_6} + {_7P_5}$ . Then the expression is $\frac{X}{X}$ .
Calculate the value of $_7P_6$ .
The formula for permutations is $_nP_r = \frac{n!}{(n-r)!}$ .
${_7P_6} = \frac{7!}{(7-6)!} = \frac{7!}{1!} = 7!$ $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$
Calculate the value of $_7P_5$ .
${_7P_5} = \frac{7!}{(7-5)!} = \frac{7!}{2!}$ ${_7P_5} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2!}{2!} = 7 \times 6 \times 5 \times 4 \times 3 = 2520$
Calculate the sum in the numerator (and denominator).
${_7P_6} + {_7P_5} = 5040 + 2520 = 7560$
Substitute the sum back into the expression and simplify.
$\frac{{_7P_6} + {_7P_5}}{{_7P_6} + {_7P_5}} = \frac{7560}{7560} = 1$
Since the sum 7560 is not zero, the division is valid.

The value of the expression is:
$\textbf{1}$

Answer