# Math Help - Pascals triangle!- questions

1. ## Pascals triangle!- questions

Hi i have a few questions i had trouble with, i need some help! here are the questions,

1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20

2)
Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2

anyone ?

3. ## Re: Pascals triangle!- questions

Originally Posted by Gurp925
Hi i have a few questions i had trouble with, i need some help! here are the questions,

1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20

2)
Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2

I would be tempted to make use of the fact that \displaystyle \begin{align*} {n \choose{ r} } = \frac{n!}{r! \left( n - r \right) ! } \end{align*} and seeing what we get upon simplification of your expressions.

4. ## Re: Pascals triangle!- questions

I should maybe tell you what the answer key says before i ask for help,

1) 45C20

2) 6C2

5. ## Re: Pascals triangle!- questions

And what have you tried so far?

6. ## Re: Pascals triangle!- questions

you can do it the following way

nCr= $$\frac{{n!}}{{r!\left( {n - r} \right)!}}$$

n C r+1= $$\frac{{n!}}{{(r + 1)!\left( {n - (r + 1)} \right)!}} = \frac{{n!}}{{(r + 1)r!\left( {n - (r)} \right)!/n - r}} = (n - r)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}}$$

therefore

nCr + n C r+1 = $$\frac{{n!}}{{r!\left( {n - r} \right)!}} + \frac{{n!}}{{(r + 1)!\left( {n - (r + 1)} \right)!}} = (n - r)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}} + \frac{{n!}}{{r!\left( {n - r} \right)!}} = \frac{{n!}}{{r!\left( {n - r} \right)!}}\left( {(n - r)/(r + 1) + 1} \right) = (n + 1)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}}$$=n+1/r+1 nCr

nCr + n C r+1= (n+1/r+1) nCr

7. ## Re: Pascals triangle!- questions

Originally Posted by Gurp925
Hi i have a few questions i had trouble with, i need some help! here are the questions,
1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20
2)
Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2
You are expected to know Pascals' equality: $\binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k}$