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Permutation and Combination Class 11

Manocha Academy・60 minutes read

Permutations involve arranging items in various ways using formulas like n! and nPr = n! / (n-r)!. The text covers the multiplication principle, factorials, and permutation formulas with examples to illustrate their application.

Insights

The multiplication principle states that the total number of occurrences is the product of the occurrences of each event, which is generalized for any finite number of events.
Permutations involve arranging objects where order matters, with factorials being crucial in permutation calculations, and the formula for taking N objects at a time being N factorial.
Understanding and memorizing the formulas for permutations, including cases with repeated objects or distinct objects, is essential for efficiently solving permutation questions and can be visualized using the box method for practical application.

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Recent questions

What is the multiplication principle?
The multiplication principle states that the total number of occurrences is the product of the occurrences of each event.

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Summary

00:00

"Permutations and Multiplication Principle Explained"

Permutations are the focus of the video, involving arranging items in different ways.
The multiplication principle is introduced, stating that the total number of occurrences is the product of the occurrences of each event.
In the example with three shirts and two pants, there are six possible ways to dress up, calculated by multiplying the number of shirts and pants.
The multiplication principle is generalized for any finite number of events, with the total occurrences being the product of each event's occurrences.
Adding caps to the shirts and pants example results in 12 possible ways to dress up, calculated by multiplying the number of shirts, pants, and caps.
The fundamental principle of addition is explained, stating that if two events can happen independently, the total ways are the sum of the ways each event can occur.
Selecting one monitor from 24 boys and 16 girls involves adding the number of boys and girls, resulting in 40 possible selections.
Selecting two monitors, one boy, and one girl, involves multiplying the number of boys and girls, resulting in 384 possible selections.
Permutations are illustrated using the word "rose," showing that 24 arrangements are possible without repetition.
Allowing repetition in the word "rose" increases the possible arrangements to 256, showcasing the impact of repetition on permutations.

16:26

Telephone Number Permutations Using Boxes

In the past, telephone numbers were five or six digits long, but now they are either nine or ten digits.
To construct a five-digit telephone number starting with 87 without repeating any digits, a method using boxes is recommended.
Representing the five-digit number as five empty boxes, the first two digits are fixed as 8 and 7.
For the third digit, there are 8 possibilities left out of the 10 digits since 8 and 7 have already been used.
The fourth digit will have 7 possibilities due to no repetitions being allowed.
Similarly, the fifth digit will have 6 possibilities left.
The total number of possible arrangements is calculated using the multiplication principle: 8 * 7 * 6 = 336 telephone numbers.
Factorials are crucial in permutation calculations, denoted by an exclamation mark (e.g., 4! = 4 * 3 * 2 * 1 = 24).
Factorials are the product of a number and the factorial of the previous number (e.g., 6! = 6 * 5!).
Permutations involve arranging objects where order matters, with permutations of all objects being n! and permutations of r objects being nPr = n! / (n-r)!.

32:06

Permutations Formula for Objects Taken at a Time

Permutations formula for taking N objects at a time is N factorial.
For N objects taken R at a time, the formula is N! / (N - R)!.
The notation for N objects taken R at a time is denoted as NPR.
Simplifying the NPR formula results in N! / (N - R)!.
The formula works for taking all N objects at a time, resulting in N factorial.
The formula also works for taking zero objects, yielding a value of 1.
When repetitions are allowed, the formula for permutations is N to the power of R.
When objects are not distinct, the formula is N! / P! where P is the number of repeated objects.
For objects with multiple repetitions, the formula is N! / (P1! * P2! * ... * Pk!) where P1, P2, etc., represent the number of each repeated object.
Generalizing the formula for permutations with repeated objects, it is N! / (P1! * P2! * ... * Pk!) where P1, P2, etc., represent the number of each repeated object.

48:49

Permutations: Formulas, Methods, and Applications

Permutations of n different objects taken r at a time without repetitions are denoted as NPR, calculated as n factorial divided by n minus r factorial.
The formula NPR works for values of r from 0 to n, with r equal to 0 resulting in one arrangement.
For the special case of taking all n objects without repetitions, it becomes npn, equal to n factorial.
When repetitions are allowed, the number of permutations increases to n to the power of r, as each of the r boxes can be filled with any of the n objects.
In cases where p objects are identical among n objects, the formula for permutations becomes n factorial divided by p factorial.
Extending this formula to include k different sets of identical objects, the final formula is n factorial divided by the product of each set's factorial.
Understanding and memorizing these formulas is crucial for efficiently solving permutation questions.
For practical application, visualizing permutations using the box method can simplify problem-solving.
In a scenario where no digit repetition is allowed, the number of possible three-digit numbers using digits 1 to 9 is 54.
Similarly, for four-digit numbers without digit repetition, the total possible permutations amount to 4,536.

01:04:32

Permutations with NPR formula and vowel grouping

In the NPR formula, R can range from 0 to n, with 0 being a special case where R can be equal to or greater than n, but typically R should be less than or equal to n. In a specific example with values of 5 and 6, only R equal to 3 is possible, not 10, as R cannot exceed n.
To find the number of different eight-letter arrangements from the word "daughter" where all vowels must be together, the total permutations are calculated as 6 factorial multiplied by 3 factorial, considering the vowels as one unit. This results in the total number of permutations where the vowels are grouped together.
For permutations where the vowels are not together, the total permutations of the word "daughter" (8 factorial) are subtracted by the permutations where the vowels are together (6 factorial multiplied by 3 factorial). Simplifying the calculation leads to the final answer of 36,000 permutations.
The concept of permutations is summarized, emphasizing the formulas used in the calculations provided. A homework question involving forming numbers between 100 and 1000 with specific digits is given for practice, encouraging application of the learned concepts.

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