Recommended problems MA 193 Fall 2015

[ 1 ] How many ways are there to form a three letter sequence using the letters a,b,c,d,e,f

( a ) without repetition of letters ?

( b ) with repetition of letters allowed ?

( c ) without repetition and containing the letter “e” once ?

Solution;

( a ) 6.5.4 = 120

( b) 63

( c ) 3.5.4 = 60

[ 2 ] how many non empty different collections can be formed from five identical apples and eight identical

oranges ?

Solution:

54-1 = 53 ( there are 6 possible values for apples and 9 possible values for oranges. Subtract the one

collection which has neither apples nor oranges since problem asked for non empty collection )

[ 3 ] How many different sequences of Heads and Tails are possible if a coin is flipped 100 times ??

Solution ; 2100

[ 4 ] How many ways are there to pick 2 different cards from a standard deck of 52 cards such that ;

( a ) the first card is an Ace and second card is NOT a Queen ?

( b ) the first card is a spade and second card is NOT a Queen ? ( beware the Queen of Spades )

Solution;

( a ) 4. 47 = 188

( b ) (1.48) + ( 12. 47 ) = 612 ( break up into two mutually exclusive events. First card picked is a Queen

of spade or Not )

[ 5 ]

( a ) how many 4 letter words ( sequence of letters with repetition ) are there in which the first and last

letters are

vowels ( ie: a,e,io,u ) ?

( b ) what if vowels appear only (if at all ) in the first and last letter ?

Solution;

( a ) 52 . 262

( b ) 262 . 212 ( the middle two letters must not be vowels so 212 while first and last letters

can be anything )

[ 6 ] how many different 8 digit binary sequences are there with six 1’s and two 0’s ??

Solution

[ 7 ] A coin is flipped 15 times. How many different sequences of 15 Heads and Tails are possible in which

the third Tail appears on the twelfth flip ?

solution;

choose 2 Tails in the first 11 flips followed by Tail on 12th flip. Then remainder can be

anything which is 23

[ 8] there are 10 white balls and 5 red balls in a box. How many ways can you select 3 white and 2 red balls ?

What is the probability that this will occur ?

Solution;

, probability is

[ 9 ] How many ways are there to arrange the letters REPLETE with no consecutive E’s ? ( ie; cannot

have any E’s together ) what is the probability that this will occur ?

Solution;

consider the letters - R – P – L – T - wherever there is a dash you can insert letter E. This

will

Result in “word” that does NOT have any consecutive E’s. Once you have inserted the E’s, the other letters

RPLT can be arranged in 4! Ways. Probability is just this divided by total number of ways to arrange 7

letters of which there are 3 of a kind. So denominator is so probability is

[ 10 ]

( a ) what is the coefficient of x40 in ( 1 + 2x )50 ?

( b ) what is ?

( c ) verify

( d ) find the coefficient of

( e ) find the coefficient of in

( f ) find the coefficient of in

Solution

( a ) binomial expansion so

( b ) this is binomial expansion of ( 1 + 2 )n therefore sum is 3n

( c )

( d )

( e )

( f )

[ 11 ]

( a ) show that

( b ) evaluate

( c ) evaluate

( d ) evaluate

( e ) evaluate

Solution

( a )

And from lecture notes

=

( b )

If you expand this you get

( c ) by #5 on handout of binomial

identity

( d )

( e )

[ 12 ] In how many ways can 10 identical candies be distributed among five children if

( a ) there are no restrictions ?

( b ) each child gets atleast one candy ?

( c ) the oldest child gets atleast two candy ?

Solution:

( a )

( b )

( c )

[ 13 ] How many ways can 20 coins be selected from four large containers filled with pennies, nickels,

dimes and quarters ? ( each container has only one type of coin )

solution;

[ 14 ]

An ice cream store has 31 different flavors of ice cream available. In how many ways can you order a dozen

ice ream cones if

( a ) you don’t want the same flavor more than once ?

( b ) a flavor may be ordered as many as 12 times ?

( c ) a flavor may be ordered as many as 11 times ?

Solution:

( a )

( b )

( c )

[ 15 ] How many ways can you toss 100 (identical) dice so that at least 3 of each type of face shows up ?

Solution

[ 16]

Two n-digit integers (leading zeros allowed) are considered equivalent if one is rearrangement of the other.

( for example 12033 , 20331, 01332 are considered equivalent five digit integers )

( a ) how many five digit integers are considered not equivalent ?

( b ) If the integers 1,3 and 7 appear at most once, how many non equivalent five digit integers are there ?

Solutions;

( a ) ie: the ten integers are ten distinct “donuts”. You are selecting 5 with repetion

allowed

( b )

ie: drop all three “special” digits (1,3 and 7) OR include one of the 3 digits OR include exactly two of the

special digits OR include all 3 of the special digits

[ 17] How many ways are there to place 12 marbles into 5 distinct jars if;

( a ) the marbles are all identical ?

( b ) the marbles are all distinct ?

Solutions

( a )

( b )

[ 18 ]

( a ) How many times is the PRINT statement executed in following program segment ?

FOR I = 1 to 20 do FOR J= 1 to I do FOR K= 1 to J do FOR M= 1 to K do PRINT ( “ I love math” )

( b )

What is the value of the “counter” after following program segment ?

( all variables I, J, N are integers )

COUNTER = 0

FOR I = 1 TO N DO

FOR J=1 TO I DO

COUNTER = COUNTER + 1

Solutions

( a )

( b )

[ 19 ]

You were bored so you tried tossing a coin 12 times and you got 5 Heads and 7 Tails.

In how many ways can these tosses result in

( a ) two runs of Heads and one run of Tail ?

( b ) three runs

Solutions;

( a )

( b )

[ 20 ] ( a ) How many ways are there to roll ten distinct dice so that all six faces appear ?

( b ) How many ways are there to select 5 card hand from regular deck of 52 cards

so that the hand contains at least one card from each suit ?

( c ) how many ways are there to distribute 4 distinct objects into 3 distinct boxes with at least one empty

box ?

( d ) How many different integer solutions are there to following equation ?

with

solution;

( a )

( b )

( c )

( d )

[ 21 ] Let

( a ) list all the subsets B of A such that

( b ) list all the subsets B of A such that

( c ) list all the subsets B of A such that

Solution;

( a )

( b )

( c )

[ 22 ] Let , find ;

( a )

( b ) P(A) is power set of A

( c )

( d )

( e )

( f )

solutions

( a )

( b )

( c ) A

( d )

( e )

( f )

[ 23 ]

TRUE or FALSE ?

( m ) list all possible subsets of

( n ) what is the power set of ?

( o ) what is the power set of ?

Solutions

( a ) TRUE

( b ) FALSE

( c ) TRUE

( d ) FALSE – the element is a set and does not match any of the numbers 0,1,2

( e ) FALSE – the elements of are all sets but those of are all numbers

( f ) FALSE - and are two different ways to represent the empty set. By definition no object

can be an element of an empty set and also the empty set cannot be a member of itself

( g ) TRUE – if

( h ) TRUE

( i ) TRUE

( j ) FALSE -

( k ) FALSE - because

( l ) FALSE – because

( m ) list of all possible subsets are :

( n ) power set is

( o ) the only subsets of is so power set is

[ 24 ] is following true or false ?

A ( B C ) = ( A B ) C

Solution - TRUE

[ 25 ] For A = { 1,2,3,4,5,6,7 }

Determine the number of

( a ) subsets of A

( b ) non empty subsets of A

( c ) proper subsets of A

( d ) nonempty proper subsets of A

( e ) subsets of A containing 3 elements

( f ) subsets of A containing 1 and 2

( g ) subsets of A containing 5 elements, including elements 1 and 2

( h ) subsets of A with an even number of elements

( i ) subsets of A with an odd number of elements

Solution

( a )

( b )

( c )

( d )

( e )

( f )

( g )

( h )

( i )

[ 26 ] For determine ;

( a )

( b )

( c )

Solution;

( a )

( b )

( c )

[ 27 ] Let ,

find ;

( a ) ( symmetric difference )

( b )

( c ) ( A – B ) – C

( d ) A – ( B - C )

Solutions;

Elements in each set are

Natural numbers

Integers

( a )

( b )

( c )

( d )

[ 28 ] For determine ;

( a )

( b ) number of relations from A to B

( c ) number of relations on A

( d ) number of relations from A to B that contains ( 1,2 ) and ( 1,5 )

( e ) number of relations from A to B that contain exactly 5 ordered pairs

Solution;

( a )

( b ) there are relations from A to B

( c ) there are relations on A

( d ) there are relations from A to B that contain

( e )

[ 29 ] For determine the number of relations on A that are;

( a ) reflexive

( b ) symmetric

( c ) reflexive and symmetric

( d ) reflexive and contains ( b, c )

Solution;

( a ) ( b ) ( c ) ( d )

[ 30 ] For Determine whether reflexive, symmetric, antisymmetric, transitive

( a ) { (1,1) , (2,2) , (3,3), (4,4) , (1,2) , (2,1), (2,3) , (3,2) }

( b ) { (1,1) , (2,2) , (3,3), (4,4) , (1,2) }

( c ) { (1,1) , (2,2) , (2,1 ), (1,2) }

Solution;

( a ) Reflexive and symmetric but not transitive and not antisymmetric

( b ) Reflexive ,antisymmetric and transitive but not symmetric

( c ) transitive and symmetric but not reflexive and not antisymmetric

[ 50 ] Let A = { 1, 2, 3, 4, 5, 6, 7 }

How many symmetric relations on A contain exactly four ordered pairs ?

Solutions

[ 51 ] For each of following relations, determine whether the relation is reflexive, symmetric, antisymmetric

or transitive

( a ) is the relation on ( positive integers ) where if a divides b

( b ) is the relation on ( integers ) where if is odd

Solution;

( a ) its reflexive, antisymmetric and transitive

( b ) symmetric

[ 31 ] determine whether following relations on the integers is ;

reflexive , symmetric , antisymmetric and transitive

( a ) iff x is a multiple of y

( b ) iff

( c ) iff

Solutions

( a ) Reflexive –Yes , Symmetric – No , Antisymmetric – No , Transitive -Yes

( b ) Reflexive –No , Symmetric – No , Antisymmetric – Yes , Transitive -Yes

( c ) Reflexive –No , Symmetric – Yes , Antisymmetric – No , Transitive -No

[ 32 ] If define on A by if x – y is a multiple of 3

( a ) show that is an equivalence relation

( b ) determine the equivalence classes and the partition of A induced by

Solution

( a ) show that is reflexive, symmetric and transitive

( b ) , ,

[ 33 ] let determine the number of relations on A that are

( a ) reflexive

( b ) symmetric

( c ) reflexive and symmetric

( d ) reflexive and contains ( x,y )

( e ) symmetric and contains ( x,y)

( f ) antisymmetric

( g ) antisymmetric and contains ( x,y )

( h ) symmetric and antisymmetric

( i ) reflexive, symmetric and antisymmetric

Solutions;

( a ) 212

( b ) 24 26

( c ) 26

( d ) 211

( e ) 24 25

( f ) 2436

( g ) 2435

( h ) 24

( i ) 1

[ 34 ] let

( a ) how many functions are there from A to B ?

how many of these are one-to-one ?, how many are onto ?

( b ) how many functions are there from B to A ?

how many of these are one-to-one ?, how many are onto ?

solutions;

( a ) 64 functions , are one-to-one and none are onto

( b ) 46 functions , none are one-to-one , are onto. S ( 6,4) is stirling’s number

[ 35 ] How many seven letter passcode using letters { a ,b, c , d} have atleast one occurrence of each of the

four letters ?

Solution

solution: let represent the position of seven letters used for passcode then

are the four letters used

The number of surjective functions A → B is = 8400

[ 36 ] consider integer 30,030 which can be factored as

30,030 = 2 x 3 x 5 x 7 x 11 x 13 how many unordered two factor factorization of 30,030 are

possible ?

One example is ( 2 x 3 x 5 ) ( 7 x 11 x 13 ) = 30 x 1001

Solution:

This is equivalent to distributing six distinct objects into two indistinguishable boxes with no box empty

Thus there are

[ 37 ] Let

Determine number of functions where

( a )

( b )

Solution

( a )

( b )

[ 38 ] suppose we have 7 different colored balls and four empty containers labeled I,II,III and IV

( a ) in how many ways can we distribute the balls so that no container is empty ?

( b ) in this collection of 7 different colored balls, one of them is say blue. In how many ways can we

Distribute the balls so that no container is left empty and the blue ball is in container II ?

( c ) If we now remove the numbers from the four containers so they are no longer distinguishable, in how

many ways can we distribute the 7 colored balls among the 4 identical containers with some containers

possibly empty

Solution

( a )

( b )

( c )

[ 39 ] ** Note** this is problem [34] revisited using Stirling number notation

( a ) How many ways are there to roll ten distinct dice so that all six faces appear ?

( b ) How many ways are there to select 5 card hand from regular deck of 52 cards

so that the hand contains at least one card from each suit ?

( c ) how many ways are there to distribute 4 distinct objects into 3 distinct boxes with at least one empty

box ?

Solution;

( a )

( b )

( c )