How many numbers can be formed from digits 1, 3, 5, 9 if repetition of digits is not allowed?
VerifiedHint: The numbers can be 1 digit, 2 digits, 3 digits, and 4 digits as well. The number of 1 digit numbers that can be formed by using the digits 1, 3, 5, and 9 are 4. For a 2 digit number, as we have 4 intakes and 3 intakes at 1st place and 2nd place respectively. So, the number of 2 digit numbers is 12. For 3 digit numbers, as we have 4 intakes, 3 intakes, and 2 intakes at 1st place, 2nd place, and 3rd place respectively. So, the number of 3 digit numbers is 24. For 4 digit numbers, as we have 4 intakes, 3 intakes, 2 intakes, and 1 intake at 1st place, 2nd place, 3rd, and 4th place respectively. So, the number of 4 digit numbers is 24. Now, the total numbers that can be formed are the summation of 1 digit numbers, 2 digit numbers, 3 digit numbers, and 4 digit numbers.
Complete step-by-step answer:
Here we have to find the number of numbers that can be formed from digits 1, 3, 5, and 9. The numbers can have 1 digit, 2 digits, 3 digits, and 4 digits as well. So, we have to find the numbers step by step. Let us continue with 1 digit number.
As we have 4 digits, so the number of single-digit numbers is 4.
Now, we have to find the number of 2 digit numbers. For the first place, we can take any of the four digits given. But we cannot take any of the 4 digits given in the second place because repetition is not allowed here. So, we have 3 intakes for second place. Let us understand with a diagram.
The total number of 2 digit numbers \[=4\times 3=12\] .
Now, we have to find the number of 3 digit numbers. For the first place, we can take any of the four digits given. But we cannot take any of the 4 digits given in the second place because repetition is not allowed here. So, we have 3 intakes for second place. Similarly, for the 3rd place, we have to choose any one digit of the remaining two digits. Let us understand with a diagram.
The total number of 3 digit numbers \[=4\times 3\times 2=24\] .
Now, we have to find the number of 4 digit numbers. For the first place, we can take any of the four digits given. But we cannot take any of the 4 digits given in the second place because repetition is not allowed here. So, we have 3 intakes for second place. Similarly, for the 3rd place, we have to choose any one digit of the remaining two digits and proceed with 1 intake for the 4th place. Let us understand with a diagram.
The total number of 4 digit numbers \[=4\times 3\times 2\times 1=24\] .
Total numbers formed = 1 digit numbers + 2 digit numbers + 3 digit numbers
=4+12+24+24
=64
Hence, 64 numbers can be formed.
Note:This question can also be solved using a combination method. We also know that the rearrangement of n digits is \[n!\] . To find the number of 2 digit numbers, we have to take any two of the four digits given and also rearrangement is possible. That is the number of 2 digit numbers are \[^{4}{{C}_{2}}\times 2!=12\] . Similarly, for 3 digit numbers, we can take any three of the four digits, and also rearrangement is possible. That is the number of 3 digit numbers are \[^{4}{{C}_{3}}\times 3!=24\] . Similarly, for 3 digit numbers, we can take any four of the four digits, and also rearrangement is possible. That is the number of 3 digit numbers are \[^{4}{{C}_{4}}\times 4!=24\] .
