Sample Problems – Difference between Permutations and Combinations

Question 1: In how many ways can you arrange the letters of the word ARTICLE, taking 4 letters at a time, without repetition, to form words with or without meaning?

Solution: 

Here from 7 letters of the word ARTICLE, we have to arrange any 4 letters to form different words.

So, n = 7 and r = 4. 

Using permutation formula ⁿ_rP = n! / (n – r)!    

⁴₇P = 7! / (7 – 4)! 

= 7!/3! 

= (7 × 6 × 5 × 4 × 3!) / 3! = 7 × 6 × 5 × 4 = 840

Thus there are 840 different ways in which we can arrange 4 letters out of the 7 letters of ARTICLE to form different words.

Question 2: How many 6 digit pin codes can be formed from the digits 0 to 9 if each pin code starts with 48 and no digit is repeated?

Solution:

Here arrange 6 digits from 0 to 9 but the first two digits of the pin code has been already decided (4 and 8). 

So we have to now arrange only 4 digits out of the remaining 8 digits (0, 1, 2, 3, 5, 6, 7, 9).

So, n = 8 and r = 4,  

⁸₄P = 8! / (8 – 4)! 

= 8! / 4! 

= (8 × 7 × 6 × 5 × 4!) / 4! 

= 8 × 7 × 6 × 5 

= 1680

Thus, 1680 different permutation in which 6 digit pin codes can be formed.

Question 3: Out of 10 students, 4 are to be selected for a trip. In how many ways the selection be made?

Solution:

In this question select 4 students out of given 10. So combination will be used here to find the answer.

n = 10 and r = 4,  

¹⁰₄C = 10! / 4!(10 – 4)! 

= 10! / 4!6! 

= (10 × 9 × 8 × 7 × 6!) / (4 × 3 × 2 × 1 × 6!) 

= (10 × 9 × 8 × 7)/(4 × 3 × 2 × 1) 

= 210

Thus there are 210 different ways of selecting 4 students out of 10.

Question 4: A bag contains 3 red, 5 black, and 4 blue balls. How many ways are there to take out three balls so that each of the colors is taken out?

Solution:

Here take out three balls of each colour. The order in which the balls are taken out does not matter. So use combination to find the answer.

Number of ways of selecting one red ball out of 3 red balls = ³₁C

Number of ways of selecting one black ball out of 5 back balls = ⁵₁C  

Number of ways of selecting one blue ball out of 4 blue balls = ⁴₁C

Total number of ways of selecting three balls of each colour = ³₁C × ⁵₁C × ⁴₁C                 

= 3 × 5 × 4 

= 60

Thus there are 60 ways of selecting three balls of each colour.

Difference between Permutations and Combinations: Probability is concerned with the chance or possibility that an event may occur or not occur if there are ‘n’ possibilities. Put simply, probability tells us the percentage of happening of an event. Probability can be expressed as a number from 0 to 1 or as a percentage. 

In this article, we will discuss the difference between Permutations and Combinations, with their definition, and formulas.