How many odd numbers less than 1000 can be formed by using the digits 0 3 5 7 when the repetition of digits is?

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Q. How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed.

A. $21$

Answer is correct (please provide a thorough explanation).

Unit digit nos. : $3$

Dual digit nos. : $2×3$

Three digit nos. : $2×2×3$ (as per the answer)

Three digit nos. : $1×3×3$ (as per my viewpoint)

Please help.

Text Solution

Solution : Since each number is less than 1000, required numbers are the 1-digit, 2-digit and 3-digit numbers. <br> One-digit numbers: Clearly, there are two one -digit odd numbers, namely 5 and 7, formed of the given digits. <br> Two-digit numbers: Since we are to form 2- digit odd numbers, we may put 5 or 7 at the unit's place. So, there are 2 ways of filling the unit's place. <br> Now, we cannot use 0 at the ten's place and the repetition of digits is allowed. So, we may fill up the ten's place by any of the digits 2, 5, 7. Thus, there are 3 ways of filling the ten's place. <br> Hence, the required type of 2-digit numbers `=(2xx3)=6.` <br> Three-digit numbers: To have an odd 3-digit number, we may put 5 or 7 at the unit's place. So, there are 2 ways of filling the unit's place. <br> We may fill up the ten's place by any of the digits 0, 2, 5, 7. So, there are 4 ways of filling the ten's place. <br> We cannot put 0 at the hundred's place. So, the hundred's place can be filled by any of the digits 2, 5, 7 and so it can be done in 3 ways. <br> `therefore " the required number of 3-digit numbers"= (2xx4xx3) =24.` <br> Hence, the total number of required type of numbers `=(2+6+24)= 32.`

Since the number is less than 1000, it could be a three-digit, two-digit or single-digit number.
Case I: Three-digit number:
Now, the hundred's place cannot be zero. Thus, it can be filled with three digits, i.e. 3, 5 and 7.Also, the unit's place cannot be zero. This is because it is an odd number and one digit has already been used to fill the hundred's place.

Thus, the unit's place can be filled by only 2 digits.

Case II: Two-digit number:
Now, the ten's place cannot be zero. Thus, it can be filled with three digits, i.e. 3, 5 and 7.Also, the unit's place cannot be zero. This is because it is an odd number and one digit has already been used to fill the ten's place,

Thus, the unit's place can be filled by only 2 digits.

Case III: Single-digit number:  It could be 3, 5 and 7.Total single-digit numbers that can be formed = 3

Hence, required number = 12 + 6 + 3 = 21