# Some clarifications about Pascal's rule needed.

I have attached a screenshot of the proof of Pascal's rule and highlighted parts I don't understand.

1) why $$(n-(k-1))$$ on numerator is without factorial? I understand, that $$(n-(k-1))$$ is lesser than n!, but if n is left with factorial then there will be two (n-(k-1)), wouldn't there? Or maybe it's because there is $$(n-(k-1))$$ both on denominator and numerator and the $$!$$ cancels out?

2) Second question regards k on the second expression of the sum ( it is highlighted in yellow). I guess the question is the same, because again there is no $$!$$ and the $$k$$ is only left so that later one we can get rid off $$(k-1)!$$?

3) The last question regards the $$(n-k+1)!$$ on denominator on third line. The question is again similar to 1st one, where is $$(n-k+1)$$ and $$(n-k)$$? How can we simply increase the base of a factorial? ( I am not sure about the notation here)

Thank you!

• Yes, thank you. – Ieva Brakmane Apr 3 at 10:22