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HELPPP WITH nCr !!!

Hello I'm currently self-teaching myself A-level maths but I seem to be stuck a little on pascal sequences and nCr.Could someone please give me a brief explanation??.Also what is nCr=0C1 ?If I apply the factorial formula it is
0!/1!(0-1)!=0!/1!(-1)! but there is no negative factorial.So is it undefined??
I would really appreciate your help!

Reply 1

RMNDK

Original post by irenee

Firstly, I find it cool that you're self-teaching A-Level maths. I dunno, I always find it cool when I hear people self-teach stuff.

Anyways, this is binomial expansion and it relates to the pascal triangle. Do you know how to construct it? You start with 1 and then on each row, add the two numbers that are diagonally above it; treat blanks as zero.

nCr is a function that tells us how many ways are there of choosing r elements (ignoring the order) from a set of a n.

For instance for (a + b)³ how many ways can I obtain a b² term?
Well, (a + b)³ = (a + b)(a + b)(a + b)

Now notice that there are 3 ways to get that b² term.
(a + b)(a + b)(a + b) = b²...
(a + b)(a + b)(a + b) = b²...
(a + b)(a + b)(a + b) = b²...

This relates to Pascal's Triangle. Treat the first row as 0. Look at the third row (because 3 is our exponent of the entire bracket). Now look at the 2nd term, (because 2 is our exponent of the term we want). You'll find that this term is also 3.

And if you do it via nCr, i.e. ³C₂ = 3

There is no point in doing ⁰C₁. Remember that n refers to our exponent of the bracket. If you're saying n = 0, then you're raising something to the power of 0.
What's anything to the power of 0? It's just 1, so the reason it doesn't work is because there's no other terms, it's only just 1.

As for your last point, negative factorials are undefined yes. This is because
-1! = -1 x -2 x -3 x -4 x -5 .... x - infinity.
However, it is possible to use this nCr method for negative exponents (and therefore involving negative factorials) however, I think I'll leave that until later. I hope that this sufficed as answer for now.

Reply 2

irenee

Thank you so much for your kind words and your comprehensive explanation.It was exactly what I was looking for.All is clear now thank youuuuuu :smile: