Question

The number of 4-digit numbers strictly greater than 4321 that can be formed from the digits 0, 1, 2, 3, 4, 5 allowing for repetition of digits is
A) 288
B) 300
C) 310
D) 360

Answer 1

We can consider the case where the first 3 digits are the same and the fourth digit can take the value greater than the given value. Then we can take the $1^{\text{st}}$ 2 digits the same and find the possible combinations. Then we can fix the $1^{\text{st}}$ digit and find the possible numbers and without fixing any digits. Then we can find the sum of the number of numbers in all the cases to get the required number.

Complete step by step solution:
We have the digits 0, 1, 2, 3, 4, 5.
We need to find the number of 4-digit numbers strictly greater than 4321 that can be formed using the given digits.
Case 1:
We can consider the case where $1^{\text{st}}$ 3 digits are fixed.
So, the number can be written as 432x. For this number to be greater than 4321, x can take values 2, 3, 4 or 5.
So, there are 4 possible numbers.
Case 2:
We can consider the case where 2 digits are fixed.
So, the number can be written as 43yx. For this number to be greater than 4321, y can take values 3, 4 or 5 and x can take any of the 6 digits.
So, the number of possible numbers is $3 \times 6 = 18$
Case 3:
We can consider the case where the $1^{\text{st}}$ digit is fixed.
So, the number can be written as 4zyx. For this number to be greater than 4321, z can take values 4 or 5 and x and y can take any of the 6 digits.
So, the number of possible numbers is $2 \times 6 \times 6 = 72$
Case 4:
We can consider the case where none of the digits are fixed.
So, the number can be written as uzyx. For this number to be greater than 4321, u can take the value of 5 only and x, y and z can take any of the 6 digits.
So, the number of possible numbers is $1 \times 6 \times 6 \times 6 = 216$
Now the total number of required numbers is given by the sum of the numbers in all the 4 cases.
$\Rightarrow 4 + 18 + 72 + 216 = 310$
Therefore, the number of 4-digit numbers strictly greater than 4321 that can be formed from the digits 0, 1, 2, 3, 4, 5 is 310.

So, the correct answer is option C.

Note:
We must take the cases of different numbers of digits fixed only in this order so that in each case we only get the numbers greater than the given number. In each case, the $1^{\text{st}}$ place from the right which is not fixed can take the integers greater than the given integer and the places to the right to it can take any of the given digits. We must take their product to get the number of required numbers in each case. For getting the total number, we must only take the sum in each case, not the product.