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 need to calculate the values of the combinations. Step 1: Calculate the value of _7C_3. The formula for combinations is _nC_r = (n!)/(r!(n-r)!). _7C_3 = (7!)/(3!(7-3)!) = (7!)/(3!4!) _7C_3 = (7 × 6 × 5 × 4!)/(3 × 2 × 1 × 4!) = (7 × 6 × 5)/(3 × 2 × 1) = (210)/(6) = 35 Step 2: Calculate the value of _7C_6. Using the combination formula: _7C_6 = (7!)/(6!(7-6)!) = (7!)/(6!1!) _7C_6 = (7 × 6!)/(6! × 1) = 7 Alternatively, we can use the property _nC_r = _nC_n-r, so _7C_6 = _7C_7-6 = _7C_1 = 7. Step 3: Substitute the calculated values into the given expression. The expression is _7C_3 + _7C_6_7C_3 - _7C_6. (35 + 7)/(35 - 7) Step 4: Perform the addition and subtraction in the numerator and denominator. (42)/(28) Step 5: Simplify the fraction. Both the numerator and the denominator are divisible by 14. (42 ÷ 14)/(28 ÷ 14) = (3)/(2) The value of the expression is: (3)/(2) That's 2 down. 3 left today — send the next one.