How many digit odd numbers can be formed by using the digits 1,2 3 4 5 6 when I repetition of digits is not allowed II the repetition of digits is allowed?

The given digits are 0, 1, 2, 3, 4, 5

To find the possible 3-digit odd numbers.

Repetition of digits is not allowed:

Since we need 3-digit odd numbers the unit place can be filled in 3 ways using the digits 1, 3 or 5.

Hundred’s place can be filled in 4 ways using the digits 0, 1, 2, 3, 4, 5 excluding 0 and the number placed in unit place.

Ten’s place can be filled in 4 ways using the digits 0, 1, 2, 3, 4, 5 excluding the digit placed in the hundred’s place.

Therefore, by the fundamental principle of multiplication

The number of 3 – digit odd numbers formed without repetition of digits using the digits 0, 1, 2, 3, 4, 5 is

= 4 × 4 × 3

= 48

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that repetition of the digits is not allowed.

Number of ways in which place (x) can be filled = 5                                           m = 5

Number of ways in which place (y) can be filled = 4      (

∵ Repetition is not allowed)                                           n = 4

Number of ways in which place (z) can be filled = 3      (

∵ Repetition is not allowed)                                           p = 3

∴ By fundamental principle of counting, the total number of 3 digit numbers formed                                         = m x n x p = 5 x 4 x 3 = 60.

How many three digit odd numbers can be formed by using the digits 1,2,3,4,5,6 If:  1) the repetition of digits is not allowed?

      2) the repetition of digits is allowed?

Asked by Topperlearning User | 04 Jun, 2014, 01:23: PM

For a odd number, we must have 1,3 or 5 at the unit's place.  So there are 3 ways of filling the unit's place.

1)     Since the repetition of digits is not allowed, the ten's place can be filled with any of the remaining 5 digits in 5 ways.  Now, four digits are left. So hundred's place can be filled in 4 ways.

So the required numbers = 3 x 5 x 4 = 60

2)     Since the repetition of digits is allowed, so each of the ten's and hundred's place can be filled in 6 ways.

Hence, required numbers = 3 x 6 x 6 = 108

Answered by | 04 Jun, 2014, 03:23: PM

Text Solution

Solution : For a number to be odd, we must have 1, 3 of 5 at the unit's palce. So, there are 3 ways of filling the unit's place. <br> Case (i) When the repetition of digits is not allowed: <br> In this case, after filling the unit's place, we may fill the ten's place by any of the remaining five digits. So, there are 5 ways of filling the ten's place. <br> Now, the hundred's place can be filled by any of the remaining 4 digits. So, there are 4 ways of filling the hundred's place. <br> So, by the fundamental principle of multiplication, the required number of odd numbers `=(3xx5xx4) = 60.` <br> Case (ii) When the repetition of digits is allowed: <br> Since the repetition of digits is allowed, so after filling the unit's place, we may fill the ten's place by any of the given six digits. So, there are 6 ways of filling the ten's place. Similarly, the hundred's palce can be filled by any of the given six digits. So, it can be filled in 6 ways. <br> Hence, by the fundamental principle of multiplication, the required number of odd numbers `= (3xx6xx6) = 108.`