The number of integer’s n with $100 \leqslant n \leqslant 999$and containing at most two distinct digits is:
$
{\text{a}}{\text{. }}252 \\
{\text{b}}{\text{. }}280 \\
{\text{c}}{\text{. }}324 \\
{\text{d}}{\text{. }}360 \\
$
Answer
384k+ views
Hint: - Calculate total numbers having distinct digits then subtract this from total numbers.
Total three digit numbers between $100 \leqslant n \leqslant 999$
$999 - 99 = 900$, because 999 and 100 are included.
Now, a three digit number is to be formed from the digits $0,1,2,3,4,5,6,7,8,9$
$ \bullet \bullet \bullet $
Since the left most place i.e. hundred’s place cannot have zero.
So, there are 9 ways to fill hundred’s place.
Since, we consider the number with distinct digits, therefore repletion is not allowed, so, ten’s place can be filled by 9 remaining ways.
So, ten’s place can be filled in 9 ways.
Similarly, to fill the unit's place, we have 8 digits remaining.
So, the unit's place can be filled by 8 ways.
So, the required number of ways in which three distinct digit number can be formed are
$9 \times 9 \times 8 = 648$
So, the numbers having all the distinct digits $ = 648$
Thus, the remaining numbers containing at most two distinct digits $ = $ total numbers $ - $numbers having all the distinct digits.
$ = 900 - 648 = 252$
Hence, option $a$ is correct.
Note: - In such types of questions first calculate the total numbers, then calculate the total numbers having distinct digits using the procedure which is stated above, then subtract these values we will get the required answer.
Total three digit numbers between $100 \leqslant n \leqslant 999$
$999 - 99 = 900$, because 999 and 100 are included.
Now, a three digit number is to be formed from the digits $0,1,2,3,4,5,6,7,8,9$
$ \bullet \bullet \bullet $
Since the left most place i.e. hundred’s place cannot have zero.
So, there are 9 ways to fill hundred’s place.
Since, we consider the number with distinct digits, therefore repletion is not allowed, so, ten’s place can be filled by 9 remaining ways.
So, ten’s place can be filled in 9 ways.
Similarly, to fill the unit's place, we have 8 digits remaining.
So, the unit's place can be filled by 8 ways.
So, the required number of ways in which three distinct digit number can be formed are
$9 \times 9 \times 8 = 648$
So, the numbers having all the distinct digits $ = 648$
Thus, the remaining numbers containing at most two distinct digits $ = $ total numbers $ - $numbers having all the distinct digits.
$ = 900 - 648 = 252$
Hence, option $a$ is correct.
Note: - In such types of questions first calculate the total numbers, then calculate the total numbers having distinct digits using the procedure which is stated above, then subtract these values we will get the required answer.
Recently Updated Pages
Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts
The lightest gas is A nitrogen B helium C oxygen D class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Which place is known as the tea garden of India class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Write a letter to the principal requesting him to grant class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE
