What is the third number in the 156th row of Pascal's triangle?

Question

I know that it is 154 x ___
how do I find out the number?

thanks in advance

Answer 1

let's look at the triangle for a few rows

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

let's also agree that, say, we use
row 6, that would be 1 5 10 10 5 1
notice that C(5,2) which is 10 gives us the 3rd term in row 6

so the third term in row 156 is
C(155,2)
= 155!/(2!153!) = 155(154)/2 = 11935

another way is to realize that the third column in Pascal's triangle makes up the "triangular" numbers.
(if you play billiards, think of the triangle of balls at the start, in the first row there is one ball, including the first and the second row you have 3 balls, including the first, second and third row you have 6 balls et.)
these numbers are generated by (n-1)(n-2)/2 for the third term in the nth row
so for the third term in row 156 we would have 155*154/2 just like above.

Answer 2

hmm.. 155!/(2!)(153!) doesn't work on the calculator.
So the fourth term would be 155x154x153/3?

Answer 3

you are right, most calculators cannot go over 69!

but if you break down 155!/(2!153!)
by its definition:
155! = (155)(154)153!

then 155!/(2!153!)
= (155)(154)153! / (2!153!)
= 155(154)/2

look for a key something like _nC_r

by definition that is n!/(r!(n-r)!)

_nC_r is just another notation for C(n,r)
with that key, your calculator should be able to do C(155,2)

on my calc I do :

155
2nd F
the _nC_r , (under the 5 on mine)
2
=

to get 11935

Answer 4

Yep, I learned the C button already, thank you very much for the explanation:)