How many numbers with no more than three digits can be formed using only the digits 1 through 7 with no digit used more than once in a given number?
A. 259
B. 249
C. 257
D. 252

Hint: We can make one digit number, two digits numbers and three digits numbers using given number. Now we will find all types number by multiplying the no. of numbers which we can use for each places. Then add them to calculate total number of numbers.

Complete step by step solution: To solve the given question, we will find the number of numbers of one digit number, two digits numbers and three digits numbers using given number.
We have 7 numbers that are 1,2, 3, 4, 5, 6, 7.
Case 1: One digit number
We can make 7 numbers of one digit numbers by using the given number.
Case 2: Two digits number
We have 7 numbers. If we use a number in unit place then we cannot use that number in ten place.
So we have 6 numbers to fill up in ten place.
Thus total number of two digits number is \[7 \times 6 = 42\]
Case 3: Three digits number
We have 7 numbers. If we use a number in unit place then we cannot use that number in ten place and hundred place.
So we have 6 numbers to fill up in ten place.
Now we have 5 numbers to fill up in hundred place, as repeating number is not allowed.
Thus total number of three digits number is \[7 \times 6 \times 5 = 210\]
The total number of numbers with no more than three digits can be formed is 7 + 42 + 210 = 259

Option ‘A’ is correct

Note: Sometimes students forgot that the repetition is not allowed. It means we cannot use same number in two different places in a number. We can use a number only one time.