*Permutations & Combinations

Ms. Suchi Patel*Permutations & CombinationsMs. Suchi PatelPermutation A permutation is defined as a possible selection of a certain number of objects taken from a group with regard to order.In permutation order does matter. It is an orderly arrangement of objects. nPr = n! (n r) !Where, n ! = n x (n-1) x(n-2) x (n 3). If r = n, nPn = n ! ; Also 0 ! = 1nPr means the number of permutations of n things taking r things at a time.Ms. Suchi Patel2Ms. Suchi PatelCombination A combination is defined as a possible selection of a certain number of objects taken from a group without regard to order. In combination order doesnt matterIt is selection of objects without order. n C r = n! (n r) ! r ! If r = n , nCn = 1

Ms. Suchi Patel3In the Active Dog competition, a blue ribbon will be awarded to the most Active dog, a red ribbon to the second most Active dog, and a yellow ribbon to the third most Active dog. If there are 8 dogs in the competition, how many different ways can the ribbons be awarded? Brandon wants to take 3 dogs with him on his morning walk. If Brandon owns 8 dogs, how many different groups of dogs can he choose for his walk?Ms. Suchi Patel4Ms. Suchi PatelExamples of Permutations:1) There are three boys A, B & C. Arrange them in a row, two at a time.2) There are three different books Physics, Chemistry & Mathematics. Arrange them in a row taking all the three at a time.3) In how many ways can 5 students be seated on a bench?4) Find the number of permutations of the letters of the word MOBILE5) Find the number of arrangements of the letters of the word ARUN. How many of these begin with A?Ms. Suchi Patel5Ms. Suchi Patel6)In how many ways can 7 different books be arranged in a shelf? In how many ways can we arrange three particular books so that they are always together?Ms. Suchi Patel6Ms. Suchi PatelExamples of Combinations:1) Find the number of combinations of the letters of the word CAKE.2) A box contains 5 Blue & 4 Red marbles. In how many ways can 4 marbles be drawn so as to include 2 red marbles?3) In how many ways can 4 people be selected from a group of 6 among which ashok is one?4) From 8 gentlemen & 4 ladies, a committee of 6 is to be formed. In how many ways can this be done so that the committee contains exactly 2 ladies?5) There are 6 bowlers & 9 batsmen in a cricket club. In how many ways can a team of 11 be selected so that the team contains atleast 4 bowlersMs. Suchi Patel7Ms. Suchi Patel6) There are 8 members in a club, of which A & B are two members. A committee of 5 is to be formed. How many of these will include A & exclude B? Ms. Suchi Patel8Ms. Suchi PatelExamplesThere are 20 people who work in an office together. Four of these people are selected to go to the same conference together. How many such selections are possible?Suppose that 4 people are to be randomly chosen out of 10 people who agreed to be interviewed in a market survey. These four people are to be assigned to four interviewers. How many possibilities are there?Nine sealed bids for oil drilling leases arrive at a regulatory agency. In how many different orders can the nine bids be opened?Ms. Suchi Patel9Ms. Suchi Patel4) In a new group of 15 employees at a restaurant, 10 are to be assigned as servers, 3 are to be assigned as hosts, & 2 are to be assigned as cashiers. In how many ways can the assignment be made? 5) A 7 - card hand is chosen from a standard 52 card deck. How many of these will have four spades & three hearts?Ms. Suchi Patel10For these 3 statements, answer whether order matters or does not matter.A.A man is redecorating his apartment and is putting up 2 paintings side by side. How many arrangements are possible if he has 6 paintings to choose from?

B.A woman must pick six apples out of a cart of 15 to purchase. How many different ways can this be done?

C.Alex must read 5 books in the next 3 days. How many ways can he do this?

D. There are 15 available toppings in a pizza restaurant. If Maria will order a 4-topping pizza, how many different pizzas could she order?

Ms. Suchi Patel11Solution:A. Order matters.If a Monet is put on one side of a wall and a Degas on the other, and then they are switched around, that is a different room design arrangement so therefore, order matters.B. Order does not matter.Since whatever 6 apples this woman chooses to purchase represent one large group without assignment, there is no order.C. Order matters.If Alex must read 5 books in 3 days, it will matter which book he read on which day. T1 on one day, T 2 on another day. The prevailing order is the days of the week.D. Order does not matter. The order of toppings on a pizza does not matter (does it matter if pepperoni is on top or mushrooms?)

Ms. Suchi Patel12