Forming Two-Digit Numbers Without Repetition
The problem asks us to find the total number of distinct two-digit numbers that can be formed using the digits {1, 2, 3, 4, 5, 6} with the condition that the digits are not repeated.
Calculating the Number of Two-Digit Numbers
We have 6 distinct digits available: {1, 2, 3, 4, 5, 6}. We need to form two-digit numbers.
- Tens Place: We can choose any of the 6 available digits for the tens place. So, there are 6 options.
- Units Place: Since repetition is not allowed, after choosing a digit for the tens place, we have 5 remaining digits left. We can choose any of these 5 digits for the units place. So, there are 5 options.
- Total Numbers: To find the total number of two-digit numbers possible without repetition, we multiply the number of options for each place. Total Numbers = (Options for Tens Place) $\times$ (Options for Units Place) Total Numbers = $6 \times 5$ Total Numbers = 30
Alternatively, this is a permutation problem where we need to arrange 2 digits out of 6. The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$. Here, $n=6$ (total digits) and $k=2$ (digits to choose). $P(6, 2) = \frac{6!}{(6-2)!} = \frac{6!}{4!} = \frac{6 \times 5 \times 4!}{4!} = 6 \times 5 = 30$.
Therefore, there are 30 unique two-digit numbers that can be formed from the digits 1, 2, 3, 4, 5, and 6 without repetition.
Important Questions from Integers
How many three digit whole numbers are there between 75 and 405?
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
