MHB How Does Modulo Arithmetic Prove a Combinatorial Identity Involving Primes?

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The discussion focuses on proving the combinatorial identity involving primes, specifically that for a prime \( p \neq 2 \) and \( 0 \leq k \leq p-1 \), the equation \( \binom{p-1}{k} \equiv (-1)^k \pmod p \) holds true. The user explores the calculation of \( \binom{p-1}{k} \) using Wilson's theorem, which states that \( (p-1)! \equiv -1 \pmod p \). They derive that \( \binom{p-1}{k} \) can be expressed in terms of factorials and ultimately simplify it to show the equivalence to \( (-1)^k \). The conversation concludes with positive affirmation of the proof's validity.
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Hey again! (Blush)

If $p \neq 2$ is a prime and $0 \leq k \leq p-1, \text{ prove that: }$
$$\binom{p-1}{k} \equiv (-1)^k \pmod p$$

I thought that I could calculate $\binom{p-1}{k}k!$ :

$$\binom{p-1}{k}k!=\frac{(p-1)!}{k!(p-(k+1))!}k!=\frac{(p-1)!}{(p-(k+1))!}$$

According to Wilson's theorem, $(p-1)! \equiv -1 \pmod p$..
But...what is equal $\mod p$ to $(p-(k+1))!$ ? (Thinking)
 
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evinda said:
Hey again! (Blush)

If $p \neq 2$ is a prime and $0 \leq k \leq p-1, \text{ prove that: }$
$$\binom{p-1}{k} \equiv (-1)^k \pmod p$$

I thought that I could calculate $\binom{p-1}{k}k!$ :

$$\binom{p-1}{k}k!=\frac{(p-1)!}{k!(p-(k+1))!}k!=\frac{(p-1)!}{(p-(k+1))!}$$

According to Wilson's theorem, $(p-1)! \equiv -1 \pmod p$..
But...what is equal $\mod p$ to $(p-(k+1))!$ ? (Thinking)

Could I show it maybe like that?

$$ \binom{p-1}{k}=\frac{(p-1)!}{k!(p-(k+1))!}=\frac{(p-k)(p-k+1) \cdots (p-1)}{k!} \equiv \frac{(p-1) \cdots (p-k+1)(p-k)}{k!} \equiv \frac{(-1) \cdots (-k+1)(-k)}{k!} \equiv \frac{(-1)^k k!}{k!} \equiv (-1)^k \pmod p $$ ?
(Nerd)
 
evinda said:
Could I show it maybe like that?

$$ \binom{p-1}{k}=\frac{(p-1)!}{k!(p-(k+1))!}=\frac{(p-k)(p-k+1) \cdots (p-1)}{k!} \equiv \frac{(p-1) \cdots (p-k+1)(p-k)}{k!} \equiv \frac{(-1) \cdots (-k+1)(-k)}{k!} \equiv \frac{(-1)^k k!}{k!} \equiv (-1)^k \pmod p $$ ?
(Nerd)

Looks good! (Nod)
 
I like Serena said:
Looks good! (Nod)

Nice,thank you very much! (Happy)
 
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