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.
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Here are the solutions to the problems. Examples: Evaluate a) Evaluate 5! Step 1: Expand the factorial. 5! = 5 × 4 × 3 × 2 × 1 Step 2: Calculate the product. 5! = 120 The value is 120. b) Evaluate (8!)/(5!) Step 1: Expand the numerator until the denominator's factorial. (8!)/(5!) = (8 × 7 × 6 × 5!)/(5!) Step 2: Cancel out the common factorial. (8 × 7 × 6 × 5!)/(5!) = 8 × 7 × 6 Step 3: Calculate the product. 8 × 7 × 6 = 336 The value is 336. c) Evaluate ((10-3)!)/((9-6)!) Step 1: Simplify the expressions inside the factorials. (10-3)! = 7! (9-6)! = 3! Step 2: Substitute the simplified factorials into the expression. (7!)/(3!) = (7 × 6 × 5 × 4 × 3!)/(3!) Step 3: Cancel out the common factorial. 7 × 6 × 5 × 4 = 840 The value is 840. d) Evaluate ((n-1)!)/((n-3)!) Step 1: Expand the numerator until the denominator's factorial. ((n-1)!)/((n-3)!) = ((n-1)(n-2)(n-3)!)/((n-3)!) Step 2: Cancel out the common factorial. (n-1)(n-2) The value is (n-1)(n-2). e) Evaluate ((n+2)!)/(n+2) Step 1: Expand the numerator. ((n+2)!)/(n+2) = ((n+2)(n+1)!)/(n+2) Step 2: Cancel out the common term. (n+1)! The value is (n+1)!. f) If (1)/(7!) + (1)/(8!) = (x)/(9!), find the value of x. Step 1: Find a common denominator for the left side, which is 9!. (1)/(7!) = (1 × 8 × 9)/(7! × 8 × 9) = (72)/(9!) (1)/(8!) = (1 × 9)/(8! × 9) = (9)/(9!) Step 2: Substitute these into the equation. (72)/(9!) + (9)/(9!) = (x)/(9!) Step 3: Combine the terms on the left side. (72 + 9)/(9!) = (x)/(9!) (81)/(9!) = (x)/(9!) Step 4: Since the denominators are equal, the numerators must be equal. x = 81 The value of x is 81. Examples: Exponent of a Prime P in (n)! The exponent of a prime p in n! is given by Legendre's formula: E_p(n!) = _k=1^ (n)/(p^k) = (n)/(p) + (n)/(p^2) + (n)/(p^3) + a) Find the exponent of 5 in 50! Step 1: Apply Legendre's formula for p=5 and n=50. E_5(50!) = (50)/(5) + (50)/(5^2) + (50)/(5^3) + E_5(50!) = (50)/(5) + (50)/(25) + (50)/(125) + Step 2: Calculate the floor values. E_5(50!) = 10 + 2 + 0 + Step 3: Sum the values. E_5(50!) = 12 The exponent of 5 in 50! is 12. b) Find the exponent of 3 in 10! Step 1: Apply Legendre's formula for p=3 and n=10. E_3(10!) = (10)/(3) + (10)/(3^2) + (10)/(3^3) + E_3(10!) = (10)/(3) + (10)/(9) + (10)/(27) + Step 2: Calculate the floor values. E_3(10!) = 3 + 1 + 0 + Step 3: Sum the values. E_3(10!) = 4 The exponent of 3 in 10! is 4. c) Find the exponent of 7 in 343! Step 1: Apply Legendre's formula for p=7 and n=343. E_7(343!) = (343)/(7) + (343)/(7^2) + (343)/(7^3) + (343)/(7^4) + E_7(343!) = (343)/(7) + (343)/(49) + (343)/(343) + (343)/(2401) + Step 2: Calculate the floor values. E_7(343!) = 49 + 7 + 1 + 0 + Step 3: Sum the values. E_7(343!) = 57 The exponent of 7 in 343! is 57. d) What is the highest power of 7 that can divide 5000! without leaving a remainder? This is equivalent to finding the exponent of 7 in 5000!. Step 1: Apply Legendre's formula for p=7 and n=5000. E_7(5000!) = (5000)/(7) + (5000)/(7^2) + (5000)/(7^3) + (5000)/(7^4) + (5000)/(7^5) + E_7(5000!) = (5000)/(7) + (5000)/(49) + (5000)/(343) + (5000)/(2401) + (5000)/(16807) + Step 2: Calculate the floor values. E_7(5000!) = 714 + 102 + 14 + 2 + 0 + Step 3: Sum the values. E_7(5000!) = 832 The highest power of 7 that can divide 5000! is 7^832. The highest power is 832. 3 done, 2 left today. You're making progress.