Factorials and Permutations

The factorial of a number is the product of all positive integers up to that number. Therefore, 5! = 5 * 4 * 3 * 2 * 1 = 120.

To calculate the number of permutations, we use the formula n!, where n is the number of items. In this case, n = 5. Therefore, the number of permutations of the letters in the word 'APPLE' is 5! = 120.

The formula for calculating the number of permutations of n items taken r at a time is nPr = n! / (n - r)!, where n is the total number of items and r is the number of items taken at a time.

To calculate the number of ways to choose 3 people from a group of 10, we use the formula nCr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. In this case, n = 10 and r = 3. Therefore, the number of ways to choose 3 people from a group of 10 is 10C3 = 10! / (10 - 3)! = 120.

The main difference between a permutation and a combination is that permutations consider the order of the items, while combinations do not. For example, the permutations of the letters A, B, and C are ABC, ACB, BAC, BCA, CAB, and CBA, while the combinations of the letters A, B, and C are AB, AC, BC, and ABC.

To calculate the number of ways to choose a president, vice president, and secretary from a group of 12 members, we use the formula nPr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. In this case, n = 12 and r = 3. Therefore, the number of ways to choose a president, vice president, and secretary from a group of 12 members is 12P3 = 12! / (12 - 3)! = 1320.

To calculate the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we use the formula nPr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. In this case, n = 5 and r = 4. Therefore, the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is 5P4 = 5! / (5 - 4)! = 3600.

To calculate the probability of getting a sum of 7 when two fair dice are rolled, we need to determine the number of ways to get a sum of 7 and the total number of possible outcomes. There are 6 ways to get a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). The total number of possible outcomes is 36, since each die has 6 sides. Therefore, the probability of getting a sum of 7 when two fair dice are rolled is 6/36 = 1/6.

To calculate the number of ways to choose 5 balls from a bag containing 10 red balls, 15 blue balls, and 20 green balls if the balls are indistinguishable, we use the formula nCr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. In this case, n = 45 and r = 5. Therefore, the number of ways to choose 5 balls from the bag is 45C5 = 45! / (45 - 5)! = 2502.

To calculate the number of ways to arrange 10 employees in a row for a photograph if two particular employees must always stand next to each other, we can consider the two employees as one unit. This reduces the number of items to be arranged to 9. Therefore, the number of ways to arrange the employees is 9!, which is 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362880.

The formula for calculating the number of circular permutations of n objects is n! / 2. This is because each object can be placed in n - 1 positions, and the order of the objects does not matter.

To calculate the number of 5-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4 with repetition allowed, we use the formula n^r, where n is the number of digits and r is the number of digits in the number. In this case, n = 5 and r = 5. Therefore, the number of 5-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4 with repetition allowed is 5^5 = 125000.

To calculate the number of ways to choose a committee of 5 people from a group of 12 people if 2 particular people must be on the committee, we can first choose the 2 particular people. This can be done in 1 way. Then, we need to choose 3 more people from the remaining 10 people. This can be done in 10C3 ways. Therefore, the total number of ways to choose a committee of 5 people from a group of 12 people if 2 particular people must be on the committee is 1 * 10C3 = 1 * 120 = 924.

To calculate the number of ways to choose 3 balls from a bag containing 6 red balls, 4 blue balls, and 2 green balls if the balls are indistinguishable, we use the formula nCr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. In this case, n = 12 and r = 3. Therefore, the number of ways to choose 3 balls from the bag is 12C3 = 12! / (12 - 3)! = 12! / 9! = 12 * 11 * 10 = 1320.