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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๐๐ฎ๐๐ฌ๐ญ๐ข๐จ๐ง: Prove 20!/6!14! + 20!/7!13! = 21!/7!14!
๐๐จ๐ฅ๐ฎ๐ญ๐ข๐จ๐ง:
We need to show that the left-hand side (LHS) is equal to the right-hand side (RHS).
LHS = 20! / (6! 14!) + 20! / (7! 13!)
To add these fractions, we need a common denominator. The least common multiple of 6! 14! and 7! 13! is 7! 14! (since 7! = 7 ร 6! and 14! = 14 ร 13!).
Rewrite the first term with the common denominator:
20! / (6! 14!) = (20! ร 7) / (6! ร 7 ร 14!) = (7 ร 20!) / (7! 14!)
Rewrite the second term with the common denominator:
20! / (7! 13!) = (20! ร 14) / (7! ร 13! ร 14) = (14 ร 20!) / (7! 14!)
Now add the two terms:
LHS = (7 ร 20!) / (7! 14!) + (14 ร 20!) / (7! 14!)
LHS = (7 ร 20! + 14 ร 20!) / (7! 14!)
Factor out 20! from the numerator:
LHS = (20! ร (7 + 14)) / (7! 14!)
LHS = (20! ร 21) / (7! 14!)
Recall that 21! = 21 ร 20!
Substitute this into the numerator:
LHS = 21! / (7! 14!)
This is equal to the RHS.
Therefore, 20! / (6! 14!) + 20! / (7! 13!) = 21! / (7! 14!) is proven.