How many 3 digit number can be formed with the digits 1 2 3 4 and 5 which are divisible by 5 and in which the digits are not repeated?

For a number to be divisible by 5,

Unit’s place digit should be 0 or 5.

Case I: when unit’s place is 0

Unit’s place digit can be selected in 1 way.

10’s place digit can be selected in 5 ways.

100’s place digit can be selected in 4 ways.

1000’s place digit can be selected in 3 ways.

10000’s place digit can be selected in 2 ways.

∴ total number of numbers = 1 × 5 × 4 × 3 × 2 = 120

Case II: when unit’s place is 5

Unit’s place digit can be selected in 1 way

10000’s place should be a non-zero number

∴ It can be selected in 4 ways

1000’s place digit can be selected in 4 ways.

100’s place digit can be selected in 3 ways.

10’s place digit can be selected in 2 ways.

∴ total number of numbers = 1 × 4 × 4 × 3 × 2 = 96

∴ Required number = 120 + 96 = 216

Solution

(i) The unit place can be filled by any one of the digits 1,2,3,4 and 5. So the unit place can be filled in 5 ways. Similarly the tens place and hundreds place can be filled in 5 ways each because repetition of digits is allowed.∴ Total number of 3-digit numbers = 5×5×5=125(ii) The unit place can be filled by any one of the digits 1,2,3,4 and 5. So the unit place can be filled in 5 ways. Now four digits are left. So the tens place can be filled in 4 ways. the hundreds place can be filled in 3 ways by the remaining 3 digits because repetition of digits is not allowed.∴ Total number of 3-digit numbers = 3×4×5=60.

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

What is a Combination?

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition. In combination, you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

nPr = (n!) / (n-r)!

Here,

n = set size, the total number of items in the set

r = subset size , the number of items to be selected from the set

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter.

nCr = n!/(n−r)!r!

Here, 

n = Number of items in set

r = Number of items selected from the set

How many 3-digit even numbers can be formed by using the digits 1,2,3,4, and 5?

Solution:

If repetition is allowed  

A three digit even number is to be formed from given 5 digits 1,2,3,4,5.

Ones place can be filled by 2 or 4 since the number is to be even. So, there are 2 ways to fill ones place.

Since, repetition is allowed , so tens place can be filled by 5 ways.

Likewise, hundreds place can also be filled by 5 ways.

So, number of ways in which three digit even numbers can be formed is 5 × 5 × 2 = 50

If repetition is not allowed

A three digit even number is to be formed from given 5 digits 1,2,3,4,5.

Ones place can be filled by 2 or 4 since the number is to be even. So, there are 2 ways to fill ones place.

Since, repetition is not allowed, so tens place can be filled by 4 ways.

Similarly, hundreds place can be filled by 3 ways.

So, number of ways in which three digit even numbers can be formed is 2 × 4 × 3 = 24

Similar Questions

Question 1: How many 3 digit odd numbers can be formed by using the digits 1,2,3,4 and 5?

Solution:

If repetition is allowed  

A three digit odd number is to be formed from given 5 digits 1,2,3,4,5.

Ones place can be filled by 1, 3 or 5 since the number is to be odd. So,

there are 3 ways to fill ones place.

Since, repetition is allowed , so tens place can be filled by 5 ways.

Similarly, hundreds place can also be filled by 5 ways.

So, number of ways in which three digit odd numbers can be formed is 5×5×3=75

If repetition is not allowed

A three digit odd number is to be formed from given 5 digits 1,2,3,4,5.

Since, for the number is to be odd , so ones place can be filled by 1, 3 or 5. So,

there are 3 ways to fill ones place.

Since, repetition is not allowed , so tens place can  be filled by 4 ways.

Similarly, hundreds place can  be filled by 3 ways.

So, number of ways in which three digit odd numbers can be formed is 3×4×3 =36

Question 2: How many 4 digit even numbers can be formed by using the digits 1,2,3,4 and 5?

Solution:

If repetition is allowed  

A four digit even number is to be formed from given 5 digits 1,2,3,4,5.

Since, for the number is to be even, so ones place can be filled by 2 or 4. So, there

are 2 ways to fill ones place.

Since, repetition is allowed, so tens place can be filled by 5 ways.

Similarly, hundreds place can also be filled by 5 ways.

Similarly, thousandth place can also be filled by 5 ways

So, number of ways in which four digit even numbers can be formed is 5 × 5 × 5 × 2 = 250

If repetition is not allowed

A four digit even number is to be formed from given 5 digits 1,2,3,4,5.

Since, for the number is to be even, so ones place can be filled by 2 or 4. So,

there are 2 ways to fill ones place.

Since, repetition is not allowed, so tens place can be filled by 4 ways.

Similarly, hundreds place can be filled by 3 ways.

Similarly, thousandth place can be filled by 2 ways

So, number of ways in which four digit even numbers can be formed is 2 × 4 × 3 × 2 = 48

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