Question

How many six-digit numbers can be formed by using the digits 1, 2, 3, and in which at least one digit is different.
(a) 724
(b) 725
(c) 726
(d) 720

Answer 1

Hint: First of all, find the total numbers formed by three digits 1, 2, and 3. Now subtract the total numbers having all digits the same from total numbers to get the desired value.

Complete step-by-step answer:

In this question, we are given 3 digits 1, 2, and 3 and we have to find the number of six-digit numbers formed by these digits such that at least one digit is different. Let us consider the information given in the question. We need to find all the 6 digit numbers formed by 3 digits 1, 2, and 3.

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Out of the above 6 places, in the first place, any one of the digits can come. So we have a total of 3 choices for the first digit
3 _ _ _ _ _
Now, again for the second digit as the repetition is allowed, we have a total of 3 choices for the second digit as well. So, we get,
$\underline{3}\times \underline{3}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}$
Similarly, for all the digits, we have a total of 3 choices of digits (1, 2, and 3). So, we get,
$\underline{3}\times \underline{3}\times \underline{3}\times \underline{3}\times \underline{3}\times \underline{3}$
Hence, we get the total 6 digit numbers formed by the digits 1, 2 and 3 as
$3\times 3\times 3\times 3\times 3\times 3$
= 36
= 729
Now, out of 729 numbers, we have to find the numbers in which at least 1 digit is different. In other words, we can say that we have to find the numbers which do not have all the digits the same. So, we get,
Total numbers having at least 1 digit different = (Total numbers) – (Numbers having all the digits same)
We know that from the digits 1, 2 3, only 3 numbers can be made where all the digits would be same and that are
1, 1, 1, 1, 1, 1
2, 2, 2, 2, 2, 2
And 3, 3, 3, 3, 3, 3
So, we get, Total numbers having at least 1 digit different = 729 – 3 = 726
So, we get a total of 726 6 digit numbers formed by 1, 2, and 3 which have at least 1 digit different.
Hence, the option (c) is correct.

Note: In permutation and combination, students must note that whenever we are asked to find at least one difference, instead we should find (Total – ‘all same’) to easily solve the question like in the above solution. Students are advised to take examples of numbers like 1 2 3 3 2 1, 1 1 1 1 1 2 or 2 1 1 1 1 3, etc. to visualize the given question.