How many 5-digit numbers can be formed using (0-9)?

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Answer:

(1) : Without Repetition of digits, #27,216# numbers;

(2) : With Repetition, #90,000# numbers.

Explanation:

There are #2# Possibilities to be considered :-

Case (1) : Digits not repeated :-

Usually, we do not write the Left-most (the #1^(st)#) digit #0#, so, this digit can be any #1# out of #1,2,...,9#.

Thus, the #1^(st)# digit can be selected in #9# ways.

Coming to the selection of the #2^(nd)# digit, keeping in mind that we can not choose the digit which we have already chosen while selecting the #1^(st)# digit, we have to select it from #10-1=9# digits, & this can be done in #9# ways.

Now, for the #3^(rd)#, we have #10-2=8# choices.

For the #4^(th), 7# and, for the last #5^(th), 6# ways are there.

Finally, using the Fundamental Principle of Counting, we can form

#9xx9xx8xx7xx6=27,216# five-digit nos. without repeating any digit.

Case (2): Digits repeated :-

In this case, the #1^(st)# digit can be selected in #9# ways as in the Case (1).

For the #2^(nd)# place, we have #10# choices from #0,1,2,...,9.# So, this can be done in #10# ways.

Now repetition is allowed. So, for the selection of digits for the places from #3^(rd)# to the #5^(th)#, we again have #10# choices to choose from.

Accordingly, if repetition of digits is permitted, the five-digit numbers can be formed in:
#9*10*10*10*10=90000# ways.

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