Learn how to prove identities involving factorials. This video provides step-by-step proofs for several factorial expressions, demonstrating techniques for simplifying and manipulating factorials. Perfect for students learning combinatorics or algebra!
top of page
bottom of page
𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Prove 8!/(10×5! - 8×4!) = 40
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
We need to show that the left-hand side (LHS) is equal to the right-hand side (RHS), which is 40.
LHS = 8! / (10 × 5! - 8 × 4!)
Recall that 5! = 5 × 4! and 8! = 8 × 7 × 6 × 5 × 4!
Substitute 5! = 5 × 4! into the denominator:
Denominator = 10 × (5 × 4!) - 8 × 4!
Denominator = 50 × 4! - 8 × 4!
Factor out 4! from the denominator:
Denominator = (50 - 8) × 4!
Denominator = 42 × 4!
Now substitute 8! in terms of 4! into the LHS:
LHS = (8 × 7 × 6 × 5 × 4!) / (42 × 4!)
Cancel out 4! from the numerator and the denominator:
LHS = (8 × 7 × 6 × 5) / 42
Simplify the numerator:
8 × 7 × 6 × 5 = 56 × 30 = 1680
Now divide by the denominator:
LHS = 1680 / 42
1680 / 42 = 40
Since LHS = 40, which is equal to the RHS, the identity is proven.