 Now that you know how factorials work, you'll need to be able to simplify algebraic expressions and solve equations that use factorials in general form. For example, if: 5! = 5 · 4 · 3 · 2 · 1 then the factorial of any number 'n' can be expressed as: n! = n·(n - 1)·(n - 2)·(n - 3)·(n - 4)· ... ·1 Whatever expression is the starting point for a factorial results in an expression where each following term is one less: (n + 3)! = (n + 3)·(n + 2)·(n + 1)·n·(n - 1)·(n - 2)·(n - 3)· ... ·1 Now have a look at some numerical examples where the fractional form means we can reduce: 6! = 6·5·4·3·2·1 = 6·5· 4! 4·3·2·1 8! = 8·7·6·5·4·3·2·1 = 8·7·6· 5! 5·4·3·2·1 Can you see that the need to write out the full factorial form, top and bottom, is not necessary, because the bottom will reduce with an identical sequence on top: 7! = 7·6 = 42 5! 11! = 11 10! Here's one with algebraic terms, showing the full expansions: n! = n·(n - 1)·(n - 2)·(n - 3)· ... ·1 = n·(n - 1)· (n - 2)! (n - 2)·(n - 3)· ... ·1 The terms that are left over can often be simplified, here shown in grey. (n+3)! = (n+3)(n+2) = n2 + 5n + 6 (n+1)! (n + 1)! = (n + 1)·(n)·(n - 1) = (n + 1)·(n2 - n) = n3 - n2 + n2 - n = n3 - n (n - 2)! Equation Solving example 1: n! = 2 (n - 2)! Start by simplifying the factorial expression: n·(n - 1) = 2 Now solve the quadratic equation n2 - n = 2 n2 - n - 2 = 0 (n - 2)(n + 1) = 0 n = 2 or n = -1 Because factorials are defined for Whole numbers only, the answer -1 is not permitted. This means n = 2 is the sole solution. Check: n! = 2 (n - 2)! 2! = 2 (2 - 2)! 2·1 = 2 [0! is defined as 1] 1 2 = 2 example 2: n! = 6n (n - 3)! n·(n - 1)·(n - 2) = 6n n(n2 - 3n + 2) = 6n n3 - 3n2 + 2n = 6n n3 - 3n2 - 4n = 0 n·(n2 - 3n - 4) = 0 n·(n - 4)·(n + 1) = 0 n = 0 n = 4 n = -1 Both n = 0 and n = -1 would result in factorials of negative numbers so they are not allowed. n = 4 is the sole answer. Check: n! = 6n (n - 3)! 4! = 6(4) (4 - 3)! 4·3·2·1 = 24 1 24 = 24 example 3: 2(n + 3)! = 180 (n + 1)! 2(n + 3)(n + 2) = 180 (n + 3)(n + 2) = 90 n2 + 5n + 6 = 90 n2 + 5n - 84 = 0 (n + 12)(n - 7) = 0 n = -12 n = 7 Once again -12 won't work Solution n = 7 Check: 2(n + 3)! = 180 (n + 1)! 2(7 + 3)! = 180 (7 + 1)! 2(10)! = 180 (8)! 2(10)(9) = 180 180 = 180 |