COMBINATORICS 4

1 In how many ways can we order 5 double deep ice cream cones from 12 availableflavours if (a) repetition of flavour is allowed and the order of the scoops is taken intoaccount, (b) repetition of flavour is allowed and the order of the scoops is not taken intoaccount, (c) repetition of flavour is not allowed and the order of the scoops is taken intoaccount, (d) repetition of flavour is not allowed and the order of the scoops is not takeninto account.

2 Find the number of the distinguishable dominoes.

3 How many ways can M men and W women be seated (a) in a row (b) at a round table,so that no two women sit next to each other?

4 How many ways can C cakes be given to B boys if each boy must get at least onecake?

5 How many ways are there to buy 9 pens from among 8 pens of each of 20 varieties?

6 How many ways are there to distribute A apples and B bananas among C chlidren?

7 How many positive integers less than one billion have five 7’s?

8 How many ways are there to pair of A men with A women?

9 How many (i) non-negative (ii) positive solutions are there to the equation (a) w +x+y + z = K (b) w + x + y + z < K where K ∈ N.

10 (a) How many ways are there to pick 12 letters from 12 A’s and 12 B’s? (b) Howmany ways are there to pick 18 letters from 12 A’s and 12 B’s? (c) How many ways arethere to pick 25 letters from 12 A’s, 12 B’s and 12 C’s?

11 How many ways are there to assign 50 agents to five different countries so that eachcountry gets 10 agents?

12 How many ways are there to arrange A a’s and B b’s such that there are at least 2 b’sbetween any 2 a’s?

13 B boys are sitting at a round table. How many ways can S of these be selected sothat no adjacent boys can be chosen?

14 How many ways are there to put F distinguishable flags on P distiguishable poles if(a) no restriction (b) if each pole must have at least one flag (c) if each pole must haveminimum M flags?

15 How many ways are there to put T toys in B distinguishable boxes so that exactly Eboxes are empty?

16 (a) How many ways are there to distribute A apples among B boys such that theyoungest two should get the same number of apples? (b) How many ways are there todistribute T distinguishable toys among B boys such that the youngest two should getthe same number of toys?

17 How many arrangement of the letters in WISCONSIN have a W adjacent to an I butno two consecutive vowels?

18 How many ways are there to distribute W disinguishable white balls and R distin-guishable red balls into B distingihable boxes?

19 How many ways are there to seat 8 boys and 13 girls at a round table so that no 2boys are adjacent and each girl siting next to at least one boy?

20 How many election outcomes are there such that exactly 3 of the 5 candidates tie forthe most votes from 38 voters?

21 How many ways are there to put R indistinguishable red flags, B indistiguishableblue flags and D different flags on P distiguishable poles if (a) no restriction (b) if eachpole must have at least one flag (c) if each pole must have minimum M flags?

22 How many ways are there to put 8 (a) Distiguishable (b) Indistinguishable balls into6 distinguishable boxes if the first two boxes together have atmost 4 balls?

COMBINATORICS 4

1 In how many ways can we order 5 double deep ice cream cones from 12 availableflavours if (a) repetition of flavour is allowed and the order of the scoops is taken intoaccount, (b) repetition of flavour is allowed and the order of the scoops is not taken intoaccount, (c) repetition of flavour is not allowed and the order of the scoops is taken intoaccount, (d) repetition of flavour is not allowed and the order of the scoops is not takeninto account.

2 Find the number of the distinguishable dominoes.

3 How many ways can M men and W women be seated (a) in a row (b) at a round table,so that no two women sit next to each other?

4 How many ways can C cakes be given to B boys if each boy must get at least onecake?

5 How many ways are there to buy 9 pens from among 8 pens of each of 20 varieties?

6 How many ways are there to distribute A apples and B bananas among C chlidren?

7 How many positive integers less than one billion have five 7’s?

8 How many ways are there to pair of A men with A women?

9 How many (i) non-negative (ii) positive solutions are there to the equation (a) w +x+y + z = K (b) w + x + y + z < K where K ∈ N.

10 (a) How many ways are there to pick 12 letters from 12 A’s and 12 B’s? (b) Howmany ways are there to pick 18 letters from 12 A’s and 12 B’s? (c) How many ways arethere to pick 25 letters from 12 A’s, 12 B’s and 12 C’s?

11 How many ways are there to assign 50 agents to five different countries so that eachcountry gets 10 agents?

12 How many ways are there to arrange A a’s and B b’s such that there are at least 2 b’sbetween any 2 a’s?

13 B boys are sitting at a round table. How many ways can S of these be selected sothat no adjacent boys can be chosen?

14 How many ways are there to put F distinguishable flags on P distiguishable poles if(a) no restriction (b) if each pole must have at least one flag (c) if each pole must haveminimum M flags?

15 How many ways are there to put T toys in B distinguishable boxes so that exactly Eboxes are empty?

16 (a) How many ways are there to distribute A apples among B boys such that theyoungest two should get the same number of apples? (b) How many ways are there todistribute T distinguishable toys among B boys such that the youngest two should getthe same number of toys?

17 How many arrangement of the letters in WISCONSIN have a W adjacent to an I butno two consecutive vowels?

18 How many ways are there to distribute W disinguishable white balls and R distin-guishable red balls into B distingihable boxes?

19 How many ways are there to seat 8 boys and 13 girls at a round table so that no 2boys are adjacent and each girl siting next to at least one boy?

20 How many election outcomes are there such that exactly 3 of the 5 candidates tie forthe most votes from 38 voters?

21 How many ways are there to put R indistinguishable red flags, B indistiguishableblue flags and D different flags on P distiguishable poles if (a) no restriction (b) if eachpole must have at least one flag (c) if each pole must have minimum M flags?

22 How many ways are there to put 8 (a) Distiguishable (b) Indistinguishable balls into6 distinguishable boxes if the first two boxes together have atmost 4 balls?