Question

the number of eight digit numbers ,with the sum of the digits equal to 12 and formed by using the digits 1,2and 3 only are

Answer 1

The number of eight-digit integers is 266.

Given:

The number of eight-digit integers, with the sum of the digits equal to 12 and formed by using the digits 1, 2, and 3.

To Find:

The number of eight-digit integers.

Solution:

To find the number of eight-digit integers we will follow the following steps:

As given in question some of the numbers are 12 and formed by using 1,2 and 3.

So,

Possible ways of that eight-digit number will be :

1) Number = 11111133

2) Number = 11111223

3) Number = 11112222

So, these are three possible ways but they can also rearrange themselves so, possible permutations will be:

For a number of ways:

$\frac{total \: digits!}{number \: of \: times \: digits \: repeating!}$

$For \: number \: in \: 1) = \frac{8!}{6! \times 2!} = \frac{8 \times 7 \times6! }{6! \times 2 \times 1 } = 28$

$For \: number \: in \: 2) = \frac{8!}{4! \times 2!} = \frac{8 \times 7 \times 6 \times 5 \times4! }{4! \times 2 \times 1 } = 168$

$For \: number \: in \: 3) = \frac{8 !}{4! \times 4!} = \frac{8 \times 7 \times \times 6 \times 5 \times 4! }{4! \times 4 \times 3 \times 2 \times 1 } = 70$

And the sum of these ways is equal to the number of eight-digit integers = 28 + 168 + 70 = 266

Henceforth, the number of eight-digit integers is 266.

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