By working out the different combinations for the sum being 6,7 and 8, (1, 6, and 21) I can see this is a diagonal row in pascals triangle but I can’t work out how these would be calculated without the assistance of the triangle once the numbers start getting larger? I have tried to look at what numbers can be added together like with 8, you can have 3,1,1,1,1,1 and change the position of 3 (ie, 1,3,1,1,1,1 also counts towards the amount of ways you can get the sum of 8) and you can the have 2,2,1,1,1,1 and change the position of the two 2’s. I’m finding I tend to miss out some ways of adding numbers together as the number gets bigger so was just wondering if I’m going about this by the right method and to keep going if I know what numbers I’m meant to be getting, or if there’s another way of going about this?
In lectures, we did and example where we had 6 coins and we calculated the probability of getting x amount of heads. It was noted to us that it came from Pascal’s triangle so I am aware it’s from the combinations and permutations area but this is the only reason I looked at the triangle once I had calculated some values for this question!
Any help would be much appreciated!