Question

Binomial Coefficients (a). How many subsets with at least 5 elements does a set with 8 elements have? (b). Find the coefficient of r" in (3 - 2.0)"+3. (c). How many ways are there to walk down from the top of Pascal's Triangle and end somewhere on the number 20?

Answer 1

a) How many subset with atleast 5 elements does a set with & elements have e Let 5= {4, 92, 93, 99, 95,96,97,98) 5158 We want subsets of s with atleast 5 elements That is, we want to constrica sets with elements (number of) 5, 6, 7 and 8. Note that we can choose 5 elements from 8 elements in (&) ways. Similarly to choose subset of 6 elements out of 8 elements can be done (8) ways to choose subsets of 7 elements and stor s elements can be done ( e) and (1) respectively So, total numbers of subsets with at least 5 elements are, 8 + 7 81 8! + t + 61 3! G! 2! 8x7X6 + 8x7 + 8+1 3 x 2X1 56 te 8+ 8 + 1 93 That is there are 93 subsets with atleast 5 elements set with 8 - elements have.

k=0 (b) Coefficient of an in (3-2x)"+3 By Binomial theorem (a+b)? (2) ant gk :. (3-2)**3= (143) 3h+3+ *<213* : Coefficient (n.63) 3' (2x)" zen is of given by Hom (n+3) 3'62,"x" n! 3! :- Coufficient of this (m+3Xn+2)(n+1) *3°-2)" 3! ECS Scanned with CamScanner

20 is th 6. 8. In pascal's triangle to find number of ways to reach 20, We have to count paths to reach 20. Now () row in the triangle. At each junchon, we can choose to go left or right. The number of ways to reach 20 is re.co. Scanned with CamScanner