# Help with basic binomial coefficient

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1. Jan 19, 2017

### Catbird

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

Hello. I'm currently working my way through Lang's Basic Mathematics and cannot make sense of this question:

Show that if n is a positive integer at most equal to m, then

$${m \choose n}+{m\choose n-1}={m+1 \choose n}$$

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The answer in the textbook is given as:

1. $${m \choose n}+{m\choose n-1}={m! \over n!(m-n)!}+{m! \over (m-n+1)!(n-1)}$$

[common denominator n!(m — n + 1)!]

2. $$= {m!(m-n+1)+m!n\over n!(m-n+1)!}$$
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I omitted the rest of the answer as I understand what follows from 2.

However I don't understand how to get such denominator from 1.

Last edited by a moderator: Jan 19, 2017
2. Jan 19, 2017

### Staff: Mentor

I moved your thread to our homework section, as it is homework-like.

In (1), you can write n! as n(n-1)! and (m-n+1)! as (m-n+1)(m-n)!. Afterwards the fractions should be easy to add with the usual methods.

3. Jan 19, 2017

### Catbird

Thank you for the quick reply. This makes sense!