How can i find all of the numbers in a certain row of pascals triangle, for this example lets say the 11th row, without having to draw out the entire triangle?
In: Mathematics
> for this example lets say the 11th row, without having to draw out the entire triangle?
There is a formula using the nCr function: *n! / (n-r)! r!*, which gives you the rth entry in the nth row.
With this rule (and understanding it) it is fairly easy to work out the numbers using a simple process. Let’s use the 11th row [well, technically the 12th row, but we’ll call it the 11th and take the top row to be the 0th row, to make the numbers nicer].
The first term is always 1.
The next term is always the row number. 11
To get the next terms, we do two things; multiply by the next number down, and divide by the next number up from 1. [Technically we did that to get 11, but it is easier to just remember that one.]
So from 11, we multiply by 10 (next number down) and divide by 2 (next number up). 110 / 2 = 55.
And then we repeat. So our next number will be 55 x 9 (next number down) divide by 3 (next number up). It is usually easier to do the division first, so 9/3 = 3, 55 x 3 = 165.
And so on. 165 x 8 / 4 = 165 x 2 = 330.
330 x 7 / 5 = … this is kind of nasty, let’s change it to 330 x 7 x 2 / 10 = 66 x 7 = 60 x 7 + 6 x 7 = 420 + 42 = 462.
462 x 6 / 6 = 462. Oooh!
At this point we have reached the middle of Pascal’s Triangle. And the triangle is symmetric, so the rest of the numbers are going to be 330, 165, 55, 11, 1, giving us the complete row:
> 1, 11, 55, 165, 462, 462, 330, 165, 55, 11, 1
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