use the formula nPr to solve
how many arrangements can be made using four of the letters of the word COMBINE if no letter is to be used more than once?
The 7 letters of the word "COMBINE" are distinct, so there are 7 choices for the first letter, only 6 for the second, 5 for the third, and 4 for the fourth.
Since the choices are independent, by the rule of multplication, the number of choices is:
7*6*5*4
=?
840
To solve this problem using the formula nPr, where n is the number of items and r is the number of items to be selected, we can follow these steps:
Step 1: Determine the values of n and r.
n = number of letters in the word COMBINE, which is 7.
r = number of letters to be selected, which is 4.
Step 2: Substitute the values of n and r into the formula nPr.
nPr = n! / (n - r)!
= 7! / (7 - 4)!
= 7! / 3!
= (7 × 6 × 5 × 4 × 3 × 2 × 1) / (3 × 2 × 1)
= 840.
Therefore, there are 840 different arrangements that can be made using four of the letters of the word COMBINE where no letter is used more than once.
To solve this problem using the formula nPr, we need to first understand what the formula represents.
The formula nPr stands for "permutations of n objects taken r at a time." It calculates the number of arrangements that can be made when selecting r objects from a set of n objects, where order matters and no object can be repeated.
In this case, we are selecting 4 letters from the word COMBINE without repetition. Let's break down the steps to find the number of arrangements:
Step 1: Determine the values for n and r.
- n represents the total number of objects (letters in the word COMBINE), which is 7 in this case.
- r represents the number of objects to be selected, which is 4 in this case.
Step 2: Apply the formula nPr.
- The formula for nPr is n! / (n - r)!, where "!" denotes factorial.
- The factorial of a number means multiplying it by all positive whole numbers less than itself. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Step 3: Substitute the values into the formula.
- n = 7, r = 4
- n! = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
- (n - r)! = (7 - 4)! = 3! = 3 x 2 x 1
Step 4: Calculate.
- nPr = n! / (n - r)!
= 7! / (7 - 4)!
= (7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1)
= 7 x 6 x 5 x 4
= 840
Therefore, the number of arrangements that can be made using four letters of the word COMBINE without repetition is 840.