Question

How many three digits and four digits odd numbers greater than 300 can be formed with the digits 0, 1, 2, 3, 4 and 5?

Answer 1

Here we will first find the possible number of digits that can be placed in each digit place. Then we will multiply these possibilities to find the total numbers of 3-digit numbers greater than 300. Similarly, the possible number of digits that can be placed in each digit place for the 4 digit number. We will multiply these possibilities to find the total numbers of 4-digit numbers greater than 300. Finally, we will add the total number of 3 and 4 digits obtained to get the required answer.

Complete step-by-step answer:
The digits given are: 0, 1, 2, 3, 4, and 5
We have to form 3-digit odd numbers greater than 300.
From the given number we can’t use 0 as the first digit because then it will not be a 3-digit number.
The digits that can be used as the first digit are \left\\{ {3,4,5} \right\\}.
The digits that can be used as the second digit are \left\\{ {0,1,2,3,4,5} \right\\} here we have assumed that the digits can be repeated in the number.
Then for the third digit we can only choose the odd digit as we have to find odd numbers so the digit for third place is \left\\{ {3,5,7} \right\\}
Therefore, the total numbers of possible 3-digit numbers $= 3 \times 6 \times 3 = 54$
The possible 3-digit numbers are 54.
Now we will form 4-digit odd numbers greater than 300.
From the given number we can’t use 0 as the first digit because then it will not be a 4-digit number.
So, for first place we have an option of 5 digits i.e.\left\\{ {1,2,3,4,5} \right\\}.
For second place, we have the option of 6 digits i.e. \left\\{ {0,1,2,3,4,5} \right\\}.
For third place, we have the option of 6 digits as \left\\{ {0,1,2,3,4,5} \right\\}.
Then for the fourth digit, we can only choose the odd digit as we have to find odd numbers so the digit for third place is 3 that is \left\\{ {3,5,7} \right\\}.
Therefore, the total numbers of possible 4-digit numbers $= 5 \times 6 \times 6 \times 3 = 540$
The possible 4-digit numbers are 540.
So, the total numbers of three and four digit numbers are $= 54 + 540 = 594$
There are a total of 594 three and four digit odd numbers greater than 300 that can be formed with the digits \left\\{ {0,1,2,3,4,5} \right\\}.

Note:
As it is given in the question that the number should be odd therefore we have taken the last term in each case with only three possibilities. We have not taken 1and 2 as the first digit for the 3 digit number because then the number will be less than 300. However, we have included 1 and 2 as the first digit in a 4 digit number because any 4 digit number will be greater than 300. Also, there is no limitation on whether the term is repeated or not, so we have taken into account that the number can be repeated and solved accordingly.