Cheatsheet: Permutation Combination Probability
Theory
Suppose if Ajay and Vijay from the earlier example did not get along and could not picked together. In this case, we can consider three cases - one when Ajay is picked, one when Vijay is picked and one when neither is picked. The sum of these three cases is equal to the number of cases without condition - cases where they are picked together. Number of ways they can be picked together is 715 ways as seen earlier. Hence, number of ways in which at max only one of them is picked = 1365 - 715= 650 ways.
Formula
Selection without arrangement:
Choosing r out of n options can be represented as
Tips
If the arrangement should always begin or end with a particular item, its order is fixed. Hence the remaining n-1 items can be arranged in (n-1)! ways.
Solved Example
Try yourself: How many 3 digit number can be formed with the digits 5, 6, 2, 3, 7 and 9 which are divisible by 5 and none of its digit is repeated?
Try yourself: In how many different ways can the letter of the word ELEPHANT be arranged so that vowels always occur together?
Try yourself: A fruit basket contains 4 oranges, 5 apples and 6 mangoes. The number of ways a person make selection of fruits from among the fruits in the basket is?
Try yourself: There are 15 points in a plane out of which 6 are collinear. Find the number of lines that can be formed from 15 points.
Try yourself: In how many ways 4 Indians, 5 Africans and 7 Japanese be seated in a row so that all person of same nationality sits together
