m12pc63

SECTION 6.3 PERMUTATIONSWITH REPEATED OBJECTS

SUPPOSE YOU HAVE FOUR CARDS: A KING, A QUEEN, A JACK,AND AN ACE. HOW MANY DIFFERENT PERMUTATIONS CANYOU HAVE?

A K Q J

A K J Q

A J Q K

A Q J K

A J K Q

A J Q K

J K Q A

J K A Q

J A Q K

J Q A K

J A K Q

J Q K A

Q A J K

Q J A K

Q K J A

Q K A J

Q A K J

Q J K A

K Q J A

K Q A J

K A J Q

K J A Q

K A Q J

K J Q A

NOW SUPPOSE ONE OF THE CARDS WAS A DUPLICATE, IE: THEJACK IS NOW AN ACE. HOW WOULD THE NUMBER OFPERMUTATIONS BE AFFECTED?

All the jack’s will be come aces and some permutations will be repeats

A K Q A

A K A Q

A A Q K

A Q A K

A A K Q

A Q K A

Q A A K

Q K A A

Q A K A

K Q A A

K A A Q

K A Q A

Every 2 permutation will become one, so the numberof permutations will be divided by 2

NOW SUPPOSE THREE CARDS WERE THE SAME, IE: THE QUEEN ISALSO AN ACE. HOW WOULD THE NUMBER OF PERMUTATIONS BEAFFECTED?

A K A A

A A A K

A A K A

K A A A

All the queen’s are now aces, and every 3remaining permutation will be a repeat

So the number of permutations with 3repeats will be divided by another 3

NOW SUPPOSE ALL FOUR CARDS WERE THE SAME, HOW WOULDTHE NUMBER OF PERMUTATIONS BE AFFECTED?

All the remaining permutationswill be the same: A A A A

So ,the number of remainingpermutations will be divided

by 4, leaving only onepermutation

A A A A

If there were 4 objects and 2 of themwere the same, the number ofpermutations will be:

If 3 of them were the same, the numberof permutations will be:

If 4 of them were the same, then number of permutations will be:

Ex: Suppose there are five cards: A king, queen, jack, and 2aces. How many different permutations will there be?

PERMUTATIONS WITH SIMILAR OBJECTS:

If there are “n” objects in total, and there are:

“a” similar objects from the first group,

“b” similar objects from the next group,

“c” similar objects from another group, and so on....

the total number of permutations will be:

Ex: Suppose there are 3 kings, 2 queen’s, 1 jack, and 4 aces.How many different permutations will there be?

Total number of cards

Number of kings

Number of queens

Number of jacks

Number of aces

Total number of permutations:

EX: DETERMINE THE NUMBER OF PERMUTATIONS WITHALL THE LETTERS IN EACH OF THE FOLLOWING WORDS.

a)BOOKKEEPER

b)BOOMSHAKALAKA

Total number of letters

Number of B’s

Number of O’s

Number of K’s

Number of E’s

“P” and “R” each have only one, soyou can just neglect them

The number of permutations will be:

Number of O’s

Number of A’s

Number of K’s

Neglect the letters with no repeats

JACK IS TRAVELLING FROM HOME TO WORK. USING THE MAPPROVIDED, IF JACK CAN ONLY TRAVEL EAST AND SOUTH, HOW MANYDIFFERENT PATHS CAN HE TAKE?

Note: Jack can ONLY

Travel EAST & SOUTH

East

South

Each path is a different

Permutation of South & East

E E E S S S

S E S E E S

S S E E S E

The total number ofpermutations will be given by:

PRACTICE: GIVEN EACH OF THE FOLLOWING GRIDS, HOWMANY DIFFERENT PATHS ARE THERE FROM A TO B IF YOUCAN ONLY TRAVEL RIGHT AND DOWN?

Any path from A to B will require 4 rightmovements and 6 down movements

So the total number of different paths will be:

HOMEWORK:

P368 #1 – 6, 8 – 12