Calculating Three-Digit Numbers Without Repetition
We need to find the number of ways to form a three-digit number using the digits from 1 to 9, with the condition that each digit can be used only once (no repetition).
Applying Permutations
This is a problem of permutations because the order of the digits matters (e.g., 123 is different from 321). We are choosing 3 digits out of 9 available digits and arranging them.
- Total number of available digits ($n$): 9 (digits 1, 2, 3, 4, 5, 6, 7, 8, 9)
- Number of digits to be used ($k$): 3
The number of permutations of $n$ items taken $k$ at a time is given by the formula:
$ P(n, k) = \frac{n!}{(n-k)!} $
Step-by-Step Calculation
- Identify $n$ and $k$: Here, $n=9$ and $k=3$.
- Apply the permutation formula: $ P(9, 3) = \frac{9!}{(9-3)!} = \frac{9!}{6!} $
- Calculate the result: $ P(9, 3) = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} $ $ P(9, 3) = 9 \times 8 \times 7 $ $ P(9, 3) = 72 \times 7 $ $ P(9, 3) = 504 $
Therefore, there are 504 distinct three-digit numbers that can be formed using the digits 1 to 9 without repetition.
Important Questions from Permutations
How many words can be formed with the letters of the word 'POSTMAN', if every word begins with T and ends with M?
In how many ways can cells in a $3 \times 3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.
- Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
- The number of ways in which the letters in the word MINING can be arranged is
- A box containing 10 identical compartments has 6 red balls and 2 blue balls. If each compartment can hold only one ball, then the number of different possible arrangements are
