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Please help me on math challenge1B(c) amc8/mathcounts winter 2017-2018 counting and probability

 
 
YangYoyo的头像
Please help me on math challenge1B(c) amc8/mathcounts winter 2017-2018 counting and probability
YangYoyo - 2018年01月15日 Monday 17:54
 

How do you know if n choose m equals n choose o? For example, how do you know that 13 choose 6 equals 13 choose 7?

 
ReynosoDavid的头像
Re: Please help me on math challenge1B(c) amc8/mathcounts winter 2017-2018 counting and probability
ReynosoDavid - 2018年01月16日 Tuesday 10:22
 

For any two numbers $n$ and $k$ it is always true that $$\binom{n}{k} = \binom{n}{n-k}.$$ We can see that from the formula for $\displaystyle \binom{n}{k}$: $$\binom{n}{k} = \frac{n!}{k!(n-k)!} \quad\text{and}\quad \binom{n}{n-k} = \frac{n!}{(n-k)!(n-(n-k))!} = \frac{n!}{(n-k)!k!}.$$

In your example they are indeed equal: $$\binom{13}{6} = \frac{13!}{6!7!} = \frac{13!}{7!6!} = \binom{13}{7}.$$ Note this is true because $6+7 = 13$.