Answer: 1, 9, 15, 35, 49, 77, 91, 105, 105, 91, 77, 49, 35, 15, 9, 1
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Explanation
The idea is to always have 1's along the left-most and right-most edges. Then to generate values we add up the numbers.
Some examples
Now move to row 8. The 1 and 8 add to 9 which is placed below and between these values. Then 8+28 = 36 is placed below and between the 8 and 28. And so on. That's how we generate 1, 9, 36, 84, 126, 126, 84, 36, 9, 1
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Another way to generate the Pascal Triangle values is to use the nCr combination formula.
The formula is [tex]nCr = \frac{n!}{r!(n-r)!}[/tex] where the exclamation marks mean factorial.
For the ninth row we will have n = 9. The r values span from r = 0 to r = 9, which makes 9+1 = 10 values in total.
Here's an example calculation for n = 9 and r = 2.
[tex]n C r = \frac{n!}{r!(n-r)!}\\\\9 C 2 = \frac{9!}{2!*(9-2)!}\\\\9 C 2 = \frac{9!}{2!*7!}\\\\9 C 2 = \frac{9*8*7!}{2!*7!}\\\\ 9 C 2 = \frac{9*8}{2!}\\\\ 9 C 2 = \frac{9*8}{2*1}\\\\ 9 C 2 = \frac{72}{2}\\\\ 9 C 2 = 36\\\\[/tex]
When r = 2, we're looking at the item in slot r+1 = 2+1 = 3. Sure enough the 3rd slot for this row is indeed 36. This confirms we did things correctly with the nCr formula. I'll let you try out other values of r.
