What is the number of permutations of the letters of the word POSSIBILITY, which contains 3 I’s and 2 S’s? What is the permutation of a family of 6 sitting on a round table? A comp... What is the number of permutations of the letters of the word POSSIBILITY, which contains 3 I’s and 2 S’s? What is the permutation of a family of 6 sitting on a round table? A company codes its customers by giving each customer an eight character code. The first 3 characters are the letters A, B and C in any order and the remaining 5 are the digits 1, 2, 3, 4 and 5 also in any order. If each letter and digit can appear only once, what is the number of customers the company can code? A password of 6 digits is made of 926002. How many possible passwords are there? Read more

Steps to Solve

1. Identify the Elements for Permutation and Combination

Begin by determining the set of elements you need to consider for permutations or combinations.

2. Define the Total Count of Elements

Count the total number of letters and digits involved in the calculations. For example, if you have 5 letters and 3 digits, your total would be 8.

3. Choose the Formula for Permutations or Combinations

Decide whether you need to find permutations (order matters) or combinations (order does not matter). The formulas are:

For permutations of $n$ items taken $r$ at a time: $$ P(n, r) = \frac{n!}{(n-r)!} $$

For combinations of $n$ items taken $r$ at a time: $$ C(n, r) = \frac{n!}{r!(n-r)!} $$

4. Apply the Formula

Insert the values you found into the chosen formula. For example, if you have to calculate the permutations of 5 letters taken 3 at a time, use:

$$ P(5, 3) = \frac{5!}{(5-3)!} $$

5. Calculate the Result

Finally, calculate the result using factorials. Factorials are the product of an integer and all integers below it (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$).