Prove that the rᵗʰ term in the nᵗʰ row of Pascal's triangle is nCr.

AI Thread Summary
The discussion focuses on proving that the rᵗʰ term in the nᵗʰ row of Pascal's triangle equals nCr, defined by the formula n!/r!(n-r)!. Participants suggest using mathematical induction as a method for the proof, starting with the base case of (x + y)^1 and assuming it holds for (x + y)^n to show it applies for (x + y)^(n+1). There is also a mention of the definition of terms in Pascal's triangle, specifically the method of summing adjacent entries to generate the next row. Overall, the conversation emphasizes the use of induction and the foundational properties of Pascal's triangle in the proof.
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Prove that the rᵗʰ term in the nᵗʰ row of Pascal's triangle is nCr.



nCr formula: n!/r!(n-r)!



I've tried everything I can but I don't know how to approach this question.
 
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Have you tried induction?

You might try (x + y)^1 show that the coefficients are 1 and 1; and then assume it's true for (x+y)^n and show if that is true then it works for the row (x + y) ^(n+1).

I haven't tried that; but it would be my first try.
 
I would go with JimRoo's first suggestion, induction. You haven't said what definition you have for the terms in Pascal's triangle. I assume it's summing pairs of adjacent entries in one row to generate the next. Use that.
 

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