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!
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by Jagdish Chawla
"Statistics is the grammar of science."
Karl Pearson
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𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Prove 8! - 6×6! = 50×6!
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
We need to show that the left-hand side (LHS) is equal to the right-hand side (RHS).
LHS = 8! - 6 × 6!
Recall that 8! = 8 × 7 × 6!
Substitute this into the LHS:
LHS = (8 × 7 × 6!) - 6 × 6!
LHS = 56 × 6! - 6 × 6!
Now, factor out 6! from both terms:
LHS = (56 - 6) × 6!
LHS = 50 × 6!
This is equal to the RHS (50 × 6!).
Therefore, 8! - 6 × 6! = 50 × 6! is proven.