What is the third number in the 156th row of Pascal's triangle?
I know that it is 154 x ___
how do I find out the number?
thanks in advance
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.
hmm.. 155!/(2!)(153!) doesn't work on the calculator.
So the fourth term would be 155x154x153/3?
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 nCr
by definition that is n!/(r!(n-r)!)
nCr 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 nCr , (under the 5 on mine)
2
=
to get 11935
Yep, I learned the C button already, thank you very much for the explanation:)
To find the third number in the 156th row of Pascal's triangle, you need to know that Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. To build the triangle, start with a row containing only the number 1, and then each subsequent row is constructed by adding the corresponding values from the previous row.
To determine the specific number in question, we need to understand that each element in a row of Pascal's triangle can be calculated using the combination formula. The combination formula calculates the number of ways to choose k items from a set of n items, expressed as "n choose k" or in mathematical notation as C(n, k). In Pascal's triangle, this formula is represented as the following:
C(n, k) = n! / (k! * (n - k)!)
In this case, the third number in the 156th row can be found by calculating C(156, 2). Using the combination formula, the calculation will be:
C(156, 2) = 156! / (2! * (156 - 2)!)
Now, to evaluate this equation, we need to compute the factorials involved. The factorial of a number is the product of all the positive integers from 1 to that number.
156! can be calculated as the product of multiplying 156 by all the positive integers below it, down to 1. However, computing factorials for large numbers can be challenging and time-consuming.
To simplify the calculation, we can note that C(n, k) is equal to C(n, n-k). This means that C(156, 2) is the same as C(156, 156-2), which simplifies to C(156, 154).
Now, we calculate C(156, 154) using the combination formula:
C(156, 154) = 156! / (154! * (156 - 154)!)
Calculating this equation requires computing the factorials of 156, 154, and 2. To manage this calculation, you can leverage various mathematical software, programming languages, or online calculators that can calculate factorials and perform the division efficiently.
Once you have computed C(156, 154), the result will be the third number in the 156th row of Pascal's triangle.