This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the value of the expression, we first observe its structure. The numerator and the denominator are identical. Step 1: Recognize that the expression is of the form (X)/(X). The given expression is _7P_6 + _7P_5_7P_6 + _7P_5. Let X = _7P_6 + _7P_5. Then the expression is (X)/(X). Step 2: Calculate the value of _7P_6. The formula for permutations is _nP_r = (n!)/((n-r)!). _7P_6 = (7!)/((7-6)!) = (7!)/(1!) = 7! 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 Step 3: Calculate the value of _7P_5. _7P_5 = (7!)/((7-5)!) = (7!)/(2!) _7P_5 = (7 × 6 × 5 × 4 × 3 × 2!)/(2!) = 7 × 6 × 5 × 4 × 3 = 2520 Step 4: Calculate the sum in the numerator (and denominator). _7P_6 + _7P_5 = 5040 + 2520 = 7560 Step 5: Substitute the sum back into the expression and simplify. _7P_6 + _7P_5_7P_6 + _7P_5 = (7560)/(7560) = 1 Since the sum 7560 is not zero, the division is valid. The value of the expression is: 1 3 done, 2 left today. You're making progress.