Total numbers formation by permutation method

Question

How many numbers greater than $3400$ can be formed using first five natural numbers?

Without and with repetition.

Answer 1

Let's talk without repetition first.

All 5 digit numbers would give $^5P\ _5=5!$

And 3, 2, and 1 digit numbers are out of the question.

And in 4 digit numbers, keeping the first digit as 4 or 5 would straight-up give an acceptable number. Hence, we have $2\times^4P\ _3$(fixing the first digit and arranging the remaining 3 digits using 4 available digits).

If the first digit is 3 then the second digit must be 4 or 5. So there goes another $2\times^3P\ _2$.

The first digit cannot be 1 or 2.

So we have $$ 5P\ _5 + 2\times ^4P\ _3 + 2\times^3P\ _2 $$ $$ =120 + 2\times 24 + 2\times 6 $$ $$ =120 + 48 + 12 $$ $$ =180 $$

So 180 such numbers are possible without repetition.

As for with repetition, infinite numbers are possible. To give you an idea, the following are some such numbers. 11111,111111,1111111,11111111,...