 # Permutation and Combinations

There are various ways to arrange a certain group of data, generally without replacement, to form subsets. Since permutations and combinations are both related to how items can be arranged or selected, sometimes it can be hard to know when to use a permutation or a combination. The main difference between the two is that permutation is an ordered combination while combination is unordered.

##### NYJC Lecture Test Q1

A shop has $7$ mountain bicycles of different brands, $5$ racing bicycles of different brands and $8$ foldable bicycles of different brands on display. A cycling club wishes to buy $6$ of these $20$ bicycles.

(i)

How many different selections can be made if there must be no more than $3$ mountain bicycles and no more than $2$ of each of the other types of bicycles?



(i) How many different selections can be made if there must be no more than $3$ mountain bicycles and no more than $2$ of each of the other types of bicycles?



The cycling club decides to buy $3$ mountain bicycles, $1$ racing bicycle and $2$ foldable bicycles and park them on a bicycle rack, which has a row of $10$ bicycle lots.

(ii)

How many different arrangements can the cycle club arrange the bicycles on the bicycle rack if all the mountain bicycles are together, both the foldable bicycles are together and the empty lots are together?



(ii) How many different arrangements can the cycle club arrange the bicycles on the bicycle rack if all the mountain bicycles are together, both the foldable bicycles are together and the empty lots are together?



(iii)

How many different arrangements are there on the cycle rack if the foldable bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?



(iii) How many different arrangements are there on the cycle rack if the foldable bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?



##### Suggested Handwritten and Video Solutions      ##### 2019 NJC P2 Q6

A group of 8 people consists of 4 married couples.

(a)

The group stands in line. Find the number of different possible orders in which no two men stand next to each other.



(a) The group stands in line. Find the number of different possible orders in which no two men stand next to each other.



(b)

The group stands in a circle. Find the number of different possible orders in which each man stands next to his wife.



(b) The group stands in a circle. Find the number of different possible orders in which each man stands next to his wife.



(c)

The group forms a committee consisting of two teams of four people each. Find the number of ways that the committee can be formed such that neither team consists of only men or women.



(c) The group forms a committee consisting of two teams of four people each. Find the number of ways that the committee can be formed such that neither team consists of only men or women.



##### Suggested Handwritten and Video Solutions      ##### 2017 CJC MYE Q13 (a)

Jack needs to drive through a series of roads to get to his work place from home. The diagram below shows the map of his possible routes. He can choose to travel northwards or eastwards at each junction.

(ii)

Find the number of different routes Jack can take.

(ii) Find the number of different routes Jack can take.

(iii)

Jack received information over the radio about a traffic accident at intersection $X$ , and wishes to avoid the intersection. Find the number of different routes he can now take.

(iii) Jack received information over the radio about a traffic accident at intersection $X$ , and wishes to avoid the intersection. Find the number of different routes he can now take. ##### Suggested Handwritten and Video Solutions ##### Permutations and Combinations Q4: Letters Selected at Random

5 letters are selected at random from the 9 letters in the word CELESTIAL. Find the number of different selections if the 5 letters include at least one E and at most one L.



##### Suggested Handwritten and Video Solutions ##### 2018 NJC MYE P2 Q5

A game is played with a number of cards each which has single letter printed on it. A player has ten cards, which are lettered $A,C,E,E,E,F,L,L,W$ and $X$ respectively.

(i)

A code is any arrangement of theses ten letters, for example, $LXCELWFEEA$.

(i) A code is any arrangement of theses ten letters, for example, $LXCELWFEEA$.

(a)

Find the total number of different codes that the player can make.



(a) Find the total number of different codes that the player can make.



(b)

Find the number of codes which begin and end with $E$ and in which the two letters $L$ are consecutive.



(b) Find the number of codes which begin and end with $E$ and in which the two letters $L$ are consecutive.



(c)

Find the number of codes such that the letter $L$’s are separated by at least two of the remaining letters.



(c) Find the number of codes such that the letter $L$’s are separated by at least two of the remaining letters.



(ii)

A second player chooses three of the first player’s ten cards at random, and takes them from the first player. Calculate the probability that

(ii) A second player chooses three of the first player’s ten cards at random, and takes them from the first player. Calculate the probability that

(a)

the three cards chosen will all carry the letter $E$ or $L$,



(a) the three cards chosen will all carry the letter $E$ or $L$,



(b)

the first player’s remaining seven cards will all carry different letters.



(b) the first player’s remaining seven cards will all carry different letters.



##### Suggested Handwritten and Video Solutions      