Question

Write down all possible two digit numbers that can be formed by using the digits $ 3 $ , $ 7 $ and $ 9 $ if the repetition of the digit is allowed ?

Answer 1

Hint: In the given question we are required to find out and write all the possible number of combinations in $ 2 $ digit numbers when we are given the digits $ 3 $ , $ 7 $ and $ 9 $ if the repetition of the digit is allowed.

Complete step-by-step answer:
So, we are to create a $ 2 $ digit number. So, we have $ 3 $ digits with us to fill up the $ 2 $ place values or positions in the $ 2 $ digit number.
Hence, we have only $ 3 $ numbers to choose from for leftmost place value in the two digit number.
So, the number of options of digits for the tens digit are $ 3 $ .
Also, the number of options of digits for the ones digit are $ 3 $ .
Now, we know that all these selections or events must be consecutive as they will form a two digit number. Hence, we will multiply the ways of selecting the number for the place values and obtain our final answer.
Therefore, the number of combinations in two digit number $ = 3 \times 3 $
 $ = 9 $
So, there are $ 9 $ ways or combinations to form a two digit number using the digits $ 3 $ , $ 7 $ and $ 9 $ if repetition is allowed.
Now, we have to write down these nine combinations.
We can write these combinations in a systematic way by fixing a number at a certain spot.
So, fixing the digit $ 3 $ at tens place, we can get three two digit numbers $ 33 $ , $ 37 $ and $ 39 $ .
Similarly, fixing the digit $ 7 $ at tens place, we can get three two digit numbers $ 73 $ , $ 77 $ and $ 79 $ .
Similarly, fixing the digit $ 9 $ at tens place, we can get three two digit numbers $ 93 $ , $ 97 $ and $ 99 $ .
So, all possible two digit numbers that can be formed by using the digits $ 3 $ , $ 7 $ and $ 9 $ if the repetition of the digit is allowed are $ 33 $ , $ 37 $ , $ 39 $ , $ 73 $ , $ 77 $ , $ 79 $ , $ 93 $ , $ 97 $ and $ 99 $ .

Note: In such a problem, one must know the multiplication rule of counting as consequent events are occurring. One must take care while doing the calculations and should recheck them so as to be sure of the final answer. We must know that the first digit should not be zero before attempting such questions as then it would not be a $ 2 $ digit number. But we were not given zero as a digit so it doesn’t matter in this question.