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 _{n}C_{r }that is the sum of the following. The symmetric pattern. _{44}C_{19 }+ _{44}C_{20
2) }Find an equivalent _{n}C_{r }that is the sum of the following. This is the recursive pattern. _{5}C_{1} + _{5}C_{2 Thanks in advance.}

Re: Pascals triangle!- questions

Re: Pascals triangle!- questions

Quote:

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 _{n}C_{r }that is the sum of the following. The symmetric pattern. _{44}C_{19 }+ _{44}C_{20
2) }Find an equivalent _{n}C_{r }that is the sum of the following. This is the recursive pattern. _{5}C_{1} + _{5}C_{2 Thanks in advance.}

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

Re: Pascals triangle!- questions

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

1) 45C20

2) 6C2

Re: Pascals triangle!- questions

And what have you tried so far?

Re: Pascals triangle!- questions

you can do it the following way

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

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

therefore

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

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

Re: Pascals triangle!- questions

Quote:

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 _{n}C_{r }that is the sum of the following. The symmetric pattern. _{44}C_{19 }+ _{44}C_{20
2) }Find an equivalent _{n}C_{r }that is the sum of the following. This is the recursive pattern. _{5}C_{1} + _{5}C_{2}

You are expected to know Pascals' equality: $\displaystyle \binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k}$