Permutation and Combination

Permutation and Combination 

What's is the differences? 

Combination 

A set of objects in which position (or order) is NOT important. 
To a combination, the trio of Brittany, Alan and Greg is THE SAME AS Greg, Brittany and Alan.                        

Permutation 

There are basically two types of permutation:

1. Repetition is Allowed: such as the lock above. It could be "333".
2. No Repetition: for example the first three people in a running race. You can't be first and second.

Formula

Permutation and Combination:

Example 

1. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
A. 24400	B. 21300
C. 210	D. 25200

 

2. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
A. 47200	B. 48000
C. 42000	D. 50400

 

3. How many 3 digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9 which are divisible by 5 and none of the digits is repeated?
A. 20	B. 16
C. 8	D. 24

Answer and Explanation

Question 1
Answer: Option D
Explanation:
Number of ways of selecting 3 consonants from 7
= ⁷C₃
Number of ways of selecting 2 vowels from 4
= ⁴C₂

Number of ways of selecting 3 consonants from 7 and 2 vowels from 4
= ⁷C₃ × ⁴C₂
$=\left(\frac{7\times 6\times 5}{3\times 2\times 1}\right)\times \left(\frac{4\times 3}{2\times 1}\right)=210$


It means we can have 210 groups where each group contains total 5 letters (3 consonants and 2 vowels).

Number of ways of arranging 5 letters among themselves
$=5!=5\times 4\times 3\times 2\times 1=120$

Hence, required number of ways
$=210\times 120=25200$

$\mathrm{Question\; 2}$  
Answer: Option D
Explanation:
The word 'CORPORATION' has 11 letters. It has the vowels 'O','O','A','I','O' in it and these 5 vowels should always come together. Hence these 5 vowels can be grouped and considered as a single letter. that is, CRPRTN(OOAIO).

Hence we can assume total letters as 7. But in these 7 letters, 'R' occurs 2 times and rest of the letters are different.

Number of ways to arrange these letters
$=\frac{7!}{2!}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}=2520$


In the 5 vowels (OOAIO), 'O' occurs 3 and rest of the vowels are different.

Number of ways to arrange these vowels among themselves $=\frac{5!}{3!}=\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}=20$

Hence, required number of ways
$=2520\times 20=50400$




Question 3
Answer: Option A
Explanation:

A number is divisible by 5 if the its last digit is 0 or 5

We need to find out how many 3 digit numbers can be formed from the 6 digits $(2,3,5,6,7,9)$
which are divisible by 5.

Since the 3 digit number should be divisible by 5, we should take the digit 5 from the 6 digits(2,3,5,6,7,9) and fix it at the unit place. There is only 1 way of doing this.

1

Since the number 5 is placed at unit place, we have now five digits(2,3,6,7,9) remaining. Any of these 5 digits can be placed at tens place

5

1

Since the digit 5 is placed at unit place and another one digit is placed at tens place, we have now four digits remaining. Any of these 4 digits can be placed at hundreds place.

4

5

1

Required Number of three digit numbers
$=4\times 5\times 1=20$


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