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How many 5 digit numbers are there that are the same when the order of their digits is reversed (example: 14341)?

Last updated date: 12th Sep 2024
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Hint: By using the example given in the question, think about the number in general sense. Think about how the digits should be placed so that the number does not change after reversing its order. Then think about the possibilities of choosing any number in place of each digit.

We have to find the number of 5 digit numbers that are the same when the order of their digits is reversed.
Let us say that the five-digit number will look like the following.
$abcba$
This number must look like above because the first digit must be the same as that of the unit place digit. The second digit must be the same as that of the ten’s place digit. Then only, they would have the same position after reversing their order.
Since, the third digit, $c$ , is in the middle. Its position will not change by changing the order. Therefore, we do not need to consider any possibilities for it.
Now, let us consider the possibilities for each digit from the left hand side.
Digit one, i.e. $a$ cannot be zero. Therefore, there are nine possibilities to allocate a digit to $a$ .
Since, other digits can be zero, there are 10 possibilities to allocate a digit to both $b$ and $c$ , respectively.
Fourth and fifth digit, i.e. $b$ and $a$ will be the same as that of the first and the second digit. Therefore, there is only one possibility of allocating any digits to them.
Hence, the total possibilities of choosing a five-digit number such that it will not change after reversing its order will be the product of all the possibilities of selecting all the digits. i.e.
$\Rightarrow 9 \times 10 \times 10 \times 1 \times 1 = 900$
Therefore, number of possibilities of selecting a five digit number such that the number will not change after reversing its order is $900$
So, the correct answer is “ $900$ ”.

Note: Key point in this question is to understand that $a$ cannot be zero and hence it would have only nine possibilities. Because if $a = 0$ , then the given number will be a four digit number. Also, you need to understand that there is only one possibility to choose the units place and ten’s place digit as they must be the same as that of the first and the second number from the left.