Suppose we want to choose six letters without replacement from nine distinct letters if the order of the choice is taken into consideration how many ways can this be done
If we have nine distinct letters and we want to choose six of them without replacement, and the order of the choice matters, then we can use the permutation formula to determine the number of ways this can be done.
The permutation formula for choosing objects without replacement, where the order matters, is given by:
P(n, r) = n! / (n - r)!
Where n is the total number of objects and r is the number of objects chosen.
In this case, we have 9 distinct letters and we want to choose 6 without replacement. Plugging these values into the permutation formula, we get:
P(9, 6) = 9! / (9 - 6)!
= 9! / 3!
= (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (3 * 2 * 1)
= 9 * 8 * 7 * 6 * 5 * 4
= 604,800
Therefore, there are 604,800 ways to choose six letters without replacement from nine distinct letters when the order of the choice is taken into consideration.
To determine the number of ways to choose six letters without replacement from nine distinct letters, we can use the concept of permutations. Since the order of the choice matters, we need to use the formula for permutations.
The formula for permutations is:
P(n, r) = n! / (n - r)!
Where:
- n is the total number of objects
- r is the number of objects to be chosen
- n! denotes the factorial of n, which means multiplying all positive integers from 1 to n
In this case, we want to choose 6 letters from a set of 9 distinct letters, so we have:
n = 9 (total number of letters)
r = 6 (number of letters to be chosen)
Using the formula for permutations, we can calculate the number of ways as:
P(9, 6) = 9! / (9 - 6)!
= 9! / 3!
Calculating the factorials:
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880
3! = 3 × 2 × 1 = 6
Plugging in the values:
P(9, 6) = 362,880 / 6 = 60,480
Therefore, there are 60,480 ways to choose six letters without replacement from nine distinct letters, considering the order of the choice.