permu and comb

What is the fundamental principle of counting?

It states that if an event can occur in m ways and another in n ways, the total is m × n.

What does the addition principle state?

If two events cannot occur together, the total ways they can occur is m + n.

Define a permutation.

An arrangement of objects in a specific order.

How do you calculate permutations of n objects taken r at a time?

Use the formula nPr = n! / (n - r)!

What happens to permutations when repetition is allowed?

The number of permutations becomes n^r.

How do permutations differ when objects are not distinct?

You divide by the factorial of the counts of each indistinguishable object.

What is a combination?

A selection of objects where the order does not matter.

How is the number of combinations of n objects taken r at a time calculated?

Use the formula nCr = n! / (r!(n - r)!)

When should you use permutations instead of combinations?

Use permutations when the arrangement of objects matters.

What is the implication of nCr = nC(n - r)?

Choosing r objects from n is the same as leaving out n - r objects.

What does nCr + nC(r - 1) equal?

It equals n + 1C r, representing the addition of ways to choose.

What is the total number of ways to select 1 boy and 1 girl from 27 boys and 14 girls?

The total is 27 × 14 = 378 ways.

How many three-digit numbers have 7 in the unit's place?

There are 90 such numbers between 99 and 1000.

What is the total number of arrangements for 5 children with 2 always together?

There are 48 arrangements.

How many ways can 3 married couples be seated with spouses together?

They can be seated in 48 ways.

What is the number of ways to choose 10 questions from 6 in Part A and 7 in Part B?

There are 266 ways to choose.

How many signals can be sent using 6 flags of different colors?

The total is 1956 signals using one or more flags.

What is the maximum number of triangles formed by m points on l1, n points on l2, and k points on l3?

It is (m + n + k)C3 - mC3 - nC3 - kC3.

What is the fundamental principle of counting?

It states that if an event can occur in m ways and another in n ways, the total is m × n.

How does the addition principle differ from the multiplication principle?

The addition principle applies when events cannot occur together, summing their individual occurrences.

What defines a permutation?

It is an arrangement of objects in a specific order.

What is nP_r?

It represents the number of permutations of n objects taken r at a time.

When is repetition of objects allowed in permutations?

Repetition allows for n^r permutations when selecting r objects from n.

What is the formula for combinations?

The number of combinations of r objects from n is given by nC_r = n! / (r!(n-r)!).

When should you use permutations instead of combinations?

Use permutations when the order of arrangement matters.

What happens if two specific objects must be together in permutations?

Treat them as a single unit, reducing the total arrangement count.

How do you calculate the number of ways to select a committee from a group?

Use combinations to determine the selection without regard to order.

What is the implication of having at least one digit repeated in forming words?

It reduces the total number of unique arrangements possible.

How do you find the number of triangles formed by points on parallel lines?

Subtract combinations of collinear points from total combinations.

What is the significance of the rank of a word in permutations?

It indicates the position of the word in an ordered list of all permutations.

What is the total number of ways to arrange n distinct objects?

It is given by n! (n factorial).

How do you determine the number of ways to color a diagram with restrictions?

Use the fundamental principle of counting considering the restrictions.

What is the condition for selecting objects when at least one must be included?

Calculate total selections and subtract cases where the object is excluded.

What is the formula for the number of ways to arrange books by subject?

Multiply the arrangements of subject groups by the arrangements within each group.

How do you calculate the number of ways to choose questions with restrictions?

Consider all valid combinations that meet the selection criteria.