Question

How many three-digit prime numbers can be formed using the digits 1, 2, and 3 without repetition?

Answer 1

Step-by-step explanation:

Well 3 digit numbers formed by 1,2,3 can be repetitive or non repetitive type

If repetition of any number is allowed:

Then 3³ = 3x3x3 = 27 possibilities

Suppose xyz is 3 digit number then

For x we have 3 possibilities, for y 3 possibilities and for x again 3 possibilities if repetition is allowed so possible numbers = 3x3x3 = 27

Numbers are

111, 112, 113, 121, 122, 123, 131, 132, 133

211, 212, 213, 221, 222, 223, 231, 232, 233

311, 312, 313, 321, 322, 323, 331, 332, 333

Total = 27

If repetition of any number is not allowed:

If repetition is not allowed then

Take example of xyz

Now for x we have 3 possibilities

For y we cannot use the number already used for x so (3–1) = 2 possibilities

For z we cannot use the number already used for x and y so we left with (3–2) = 1 possibility

So total 3 digit number possibilities without repetition is 3x2x1 = 6

Numbers are

123, 132, 213, 231, 312, 321

Total = 6

We can also calculation it through permutation :

nPr = n! / (n-r)!

Now we have to form 3 digit number so r=3

We have 3 numbers available so n=3

So

3P3 = 3!/ (3–3)! = 3x2x1 / 0! = 6 /1 = 6

Or if you have required outcome = available resource as in our case both required number = 3 and available digits = 3

We can use directly 3! = 3x2x1 = 6

Hope it will help

Answer 2

Given: Three digit numbers are formed using 1,2 and 3 without repetition

To find: Number of prime numbers out of the formed numbers

Explanation: The three numbers 1,2 and 3 can be arranged in any order to form the three digit numbers.

The formula for calculating the total number of ways is

= 3!

= 3×2×1

= 6 ways

The numbers are 123,321,132,231,213 and 312.

A prime number is a number which can only 1 and the number itself as the factors.

But, here each of these numbers are divisible by atleast 3 which proves that they have more factors other than 1 and the number itself.

Therefore, 0 three digit prime numbers can be formed using 1,2 and 3 without repetition.