How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 ar

1.	How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?
Answer» Here, repetition of digits is not allowed. 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 the 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 ∴ Total number of required numbers = 120 + 96 = 216

How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?

Answer»

Here, repetition of digits is not allowed.

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 the 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

∴ Total number of required numbers

= 120 + 96 = 216