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 15!/5!10! + 15!/4!11! + 16!/5!11!
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
We need to show that the left-hand side (LHS) is equal to the right-hand side (RHS).
LHS = 15! / (5! 10!) + 15! / (4! 11!)
To add these fractions, we need a common factor in the denominators. Notice that 5! = 5 × 4! and 11! = 11 × 10!. The least common multiple of the denominators can be thought of in terms of combinations: ¹⁵C₅ + ¹⁵C₄ should relate to ¹⁶C₅.
Let's find a common denominator: 5! 11! (since 11! = 11 × 10! and we can multiply the second term's denominator by 5/5).
Rewrite the first term with the common denominator:
15! / (5! 10!) = (15! × 11) / (5! × 10! × 11) = (11 × 15!) / (5! 11!)
Rewrite the second term with the common denominator:
15! / (4! 11!) = (15! × 5) / (4! × 5 × 11!) = (5 × 15!) / (5! 11!)
Now add the two terms:
LHS = (11 × 15!) / (5! 11!) + (5 × 15!) / (5! 11!)
LHS = (11 × 15! + 5 × 15!) / (5! 11!)
LHS = (15! × (11 + 5)) / (5! 11!)
LHS = (15! × 16) / (5! 11!)
Recall that 16! = 16 × 15!
Substitute this into the numerator:
LHS = 16! / (5! 11!)
This is equal to the RHS.
Therefore, 15! / (5! 10!) + 15! / (4! 11!) = 16! / (5! 11!) is proven.