Determine the number of ways of selecting four letters, without regard for order, from the word 'parallelogram'
To determine the number of ways of selecting four letters without regard for order, we need to calculate the number of combinations.
In the word 'parallelogram', we have a total of 14 letters. We want to select four of these letters.
The formula to calculate combinations is given by nCr, where n is the total number of items and r is the number of items to be selected.
nCr = n! / (r! * (n - r)!)
Here, n! denotes the factorial of n, which is the product of all positive integers less than or equal to n.
Let's calculate the combinations for this problem:
n = 14 (total number of letters)
r = 4 (number of letters to be selected)
nCr = 14! / (4! * (14 - 4)!)
Calculating the factorials:
14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
4! = 4 * 3 * 2 * 1
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Substituting the factorials:
nCr = 14! / (4! * 10!)
Now, we can simplify the expression:
nCr = (14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
After simplifying, we get:
nCr = (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1)
Calculating the numerator and denominator separately:
Numerator = 14 * 13 * 12 * 11 = 24024
Denominator = 4 * 3 * 2 * 1 = 24
Finally, dividing the numerator by the denominator:
nCr = 24024 / 24 = 1001
Therefore, there are 1001 ways to select four letters without regard for order from the word 'parallelogram'.