Permutations & Combinations- 2 Video Lecture | Quantitative Aptitude for CA Foundation

Ans. Permutations and combinations are both methods used to count the number of possible outcomes in a given situation, but they are used in different scenarios. Permutations are used when the order of elements matters. For example, when arranging a group of people in a line, the order in which they are arranged is important. The formula for calculating permutations is nPr = n! / (n - r)!, where n is the total number of elements and r is the number of elements to be arranged. Combinations, on the other hand, are used when the order does not matter. For example, when selecting a committee from a group of people, the order in which they are selected does not matter. The formula for calculating combinations is nCr = n! / (r! * (n - r)!), where n is the total number of elements and r is the number of elements to be selected.

Ans. When repetition is allowed, the formula for calculating the number of permutations changes slightly. In this case, the formula is n^r, where n is the total number of elements and r is the number of elements to be arranged. For example, if you have 3 different colors and you want to arrange them in a row with repetition allowed, there are 3 choices for each position. So, the total number of permutations would be 3^3 = 27.

Ans. Permutations and combinations have various applications in real-life situations. Here are a few examples: - In sports, the number of possible outcomes in a tournament or league can be calculated using permutations and combinations. For example, the number of possible match-ups in a tennis tournament can be calculated using combinations. - In probability theory, permutations and combinations are used to calculate the probability of certain events occurring. For example, the probability of drawing a specific card from a deck of cards can be calculated using combinations. - In computer science, permutations and combinations are used in algorithms and data structures. For example, in a password cracker, permutations are generated to try different combinations of characters to crack a password.

Ans. In terms of sample space, the main difference between a permutation and a combination is the number of possible outcomes. In a permutation, each arrangement is considered as a separate outcome. For example, if you have 3 different colors and you want to arrange them in a row, each different arrangement is considered a separate outcome. So, the sample space for permutations would be larger compared to combinations. In a combination, the order of elements does not matter, so different arrangements of the same elements are considered as one outcome. For example, if you want to select a committee of 3 people from a group of 10, different orders of the same 3 people would be considered as one outcome. So, the sample space for combinations would be smaller compared to permutations.

Ans. Yes, permutations and combinations are used extensively in probability calculations. In probability, permutations and combinations are used to calculate the number of favorable outcomes and the total number of possible outcomes. By dividing the number of favorable outcomes by the total number of possible outcomes, the probability of an event occurring can be calculated. For example, if you want to calculate the probability of rolling a 6 on a fair six-sided die, the number of favorable outcomes (1) can be divided by the total number of possible outcomes (6) to get the probability of 1/6. In more complex probability problems, permutations and combinations are used to calculate the number of ways an event can occur and the total number of possible outcomes, allowing for the calculation of probabilities.