How many four-digit numbers can you form using ten numbers (0, 1, 2, 3

We have to make 4 digit number without repetition of digit

i.e. To make the Number, we have to fill these 4 spaces

_ _ _ _

The leftmost space can be filled by any digit except 0 hence 9 choices,

9 _ _ _

The Second from left space can be filled by any digit except one digit used i.e. in 9 ways,

9 * 9 _ _

The Third from left space can be filled by any digit except two digits used i.e. in 8 ways,

9 * 9 * 8 _

The Forth from left space can be filled by any digit except three digits used i.e. in 7 ways,

9 * 9 * 8 * 7 = 4536

ANswer: Option B

_ _ _ _

No repetition means...
Position 1: 9 possibilities (0 cannot be the first number)
Position 2: 9 possibilities (0 can be the second number)
Position 3: 8
Position 4: 7

9 x 9 x 8 x 7 = 4536

Answer is B.

Solution

Given
In this question, we are given that

• Ten digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

To find
We need to determine

• The number of 4-digit numbers we can form using these 10 digits, without repeating any digit

Approach and Working out

• We will solve this question by counting the possibilities for the digits that can be used for each place in the desired 4-digit number and then multiplying these.

o First place (thousands) can be occupied by any of the 9 digits from 1 through 9.

 These 9 digits exclude 0 since if the first digit is 0, the number formed will not be a 4-digit number.


o Second place (hundreds) can be occupied by any of the 9 digits except the one used in the first place.

 These 9 digits include 0 and eight of the digits left from 1 through 9.


o Third place (tens) can be occupied by any of the remaining 8 digits.

 These 8 digits exclude the two digits already used in the first two places.


o Fourth place (ones) can be occupied by any of the remaining 7 digits.

 These 7 digits exclude the three digits already used in the first three places.

• Total number of 4-digit numbers that can be formed = 9 × 9 × 8 × 7 = 4536

Thus, option B is the correct answer.

Correct Answer: Option B

Solution:

If the number does not have a zero digit, then we have 9P4 = 9 x 8 x 7 x 6 = 3024 such numbers.

If the number has a zero digit and the zero digit is the units digit, then we have 9P3 = 9 x 8 x 7 = 504 ways to permute the other 3 digits. Therefore, we have 504 such numbers. Similarly, if the zero digit is the tens digit or the hundreds digit, we would also have 504 numbers in each of these cases. Of course, we can’t have the zero digit as the thousands digit since a 4-digit number can’t begin with 0.

Therefore, there are a total of 3024 + 504 x 3 = 4,536 such numbers.

Alternate Solution:

For the thousands digit, there are 9 choices since 0 cannot be the thousands digit of a four-digit integer. For the hundreds digit, there are also 9 choices since we exclude the number in the thousands digit but include 0. For the tens and units digit, there are 8 and 7 choices, respectively. Thus, there are 9 x 9 x 8 x 7 = 4,536 such numbers.

Answer: B

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