Combinatorial proof: Prove the following i*["n choose i"] = n*["n-1 choose i-1"].

Show the following i*["n choose i"] = n*["n-1 choose i-1"]. (See attached file for the more mathematical version.)

This problem looks at probabilities in terms of one minus the probability of the compliment in order to solve the problem: if a die is rolled 4 times, what is the probability that a 3 comes up at least once

If a die is rolled 4 times, what is the probability that a 3 comes up at least once?

Permutations and combinations

A set of 10 flags, 5 red, 3 blue and 2 yellow are to be arranged in a line along a balcony. If flags of the same colour are INDISTINGUISHABLE, find the number of arrangements in which, 1) The three blue flags are together 2) The yellow flags are not together 3) The red flags occupy alternate positions in the line 4) If there is room for only 9 of the flags, find the total number of possible arrangements.

Seating arrangements at a dinner table

In how many ways can 6 couple be seated at a circular table if each couple is not to be separated? How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?

Permutations and combinations

Find the number of different selections of three letters which can be made from the letters of the word PARALLELOGRAM. How many of these contain the letter P?