Unit 5: Permutation, Combination and Probability - Subjective Questions

Addition principle: If an activity can be performed in ways or in mutually exclusive ways, then it can be performed in ways.

Example: A student can choose one book from 4 mathematics books or 3 physics books. The number of choices is .

Multiplication principle: If an activity consists of two successive stages that can be completed in and ways respectively, then the complete activity can be performed in ways.

Example: If there are 3 shirts and 2 trousers, the number of outfits is .

The addition principle is used for alternative cases, whereas the multiplication principle is used for successive stages.

The committee may contain 2, 3, 4, or 5 women.

Therefore, the required number of committees is

Hence, the committee can be selected in 1056 ways.

To arrange objects from distinct objects:

• The first position can be filled in ways.
• The second position can be filled in ways.
• Continuing in this manner, the th position can be filled in ways.

Thus,

Multiplying and dividing by gives

For 8 books taken 3 at a time,

Therefore, the number of arrangements is 336.

For the first digit, 0 cannot be used. Therefore, the first digit has 6 choices. The remaining four positions can be filled from the remaining 6 digits:

Thus, 2160 five-digit numbers can be formed.

For even numbers, the last digit must be one of .

Case 1: Last digit is 0

The first digit has 6 choices, and the middle three positions can be filled in ways:

Case 2: Last digit is 2, 4, or 6

The last digit has 3 choices. The first digit can be selected in 5 ways, excluding 0 and the selected last digit. The middle three positions can be filled in ways:

Total even numbers:

Hence, 1260 are even.

The word MISSISSIPPI contains 11 letters with repetitions:

• occurs 4 times.
• occurs 4 times.
• occurs 2 times.
• occurs 1 time.

Therefore, the total number of distinct arrangements is

For arrangements in which all four I's are together, consider the four I's as one block. We then arrange the objects , giving 8 objects.

The number of arrangements is

Hence, the answers are:

• Total arrangements:
• Arrangements with all I's together:

Permutation: A permutation is an arrangement in which order matters. For example, AB and BA are different arrangements.

Combination: A combination is a selection in which order does not matter. For example, AB and BA represent the same selection.

The formulas are

and

Therefore,

For selecting 4 students from 10,

Thus, 4 students can be selected in 210 ways.

For arranging objects in a line, there are arrangements. In a circular arrangement, rotations of the same arrangement are considered identical. Each circular arrangement can be rotated in ways, so we divide by :

Thus, the number of circular arrangements of distinct objects is

For 6 people,

Therefore, the 6 people can be seated in 120 circular arrangements.

First, arrange the 5 men around the circular table:

These men create 5 spaces between consecutive men. To ensure that no two women sit together, the 4 women must occupy 4 of these 5 spaces.

The number of ways to select the spaces is

The 4 women can be arranged in these selected spaces in

ways.

Hence, the total number of arrangements is

Therefore, the required number of arrangements is 2880.

A chord is determined by selecting any 2 of the 10 points. Therefore,

A triangle is determined by selecting any 3 of the 10 points. Therefore,

Hence:

• Number of chords:
• Number of triangles:

Random experiment: An experiment whose exact outcome cannot be predicted in advance, although all possible outcomes are known.

Sample space: The set of all possible outcomes of a random experiment, usually denoted by .

Event: Any subset of the sample space.

Classical probability: If an experiment has equally likely, mutually exclusive, and exhaustive outcomes, and an event contains favorable outcomes, then

The classical definition applies when:

• The number of possible outcomes is finite.
• All outcomes are equally likely.
• The outcomes are mutually exclusive.
• The outcomes are exhaustive, meaning that one of them must occur.

The probability always satisfies

Mutually exclusive events: Two events that cannot occur together. For a single die throw, getting 2 and getting 5 are mutually exclusive.

Exhaustive events: Events whose union contains all possible outcomes. In a coin toss, head and tail are exhaustive.

Independent events: The occurrence of one event does not affect the probability of the other. Results of two separate coin tosses are independent.

Dependent events: The occurrence of one event changes the probability of another. Drawing two cards without replacement gives dependent events.

Complementary events: For an event , its complement contains all outcomes not in . They satisfy

Equally likely events: Events having the same probability of occurrence. In a fair die, each face is equally likely.

The sample space is

There are 4 equally likely outcomes.

(a) Exactly one head

Favorable outcomes are and .

(b) At least one head

Favorable outcomes are .

Alternatively, .

(c) Two tails

Only is favorable.

For 4 tosses, the total number of outcomes is

Exactly 2 heads: Choose the positions of the 2 heads:

Therefore,

At least 3 heads: This includes 3 heads and 4 heads:

Thus,

No heads: There is only one outcome, :

When two dice are thrown, the total number of equally likely outcomes is

(a) Sum equal to 7: The favorable pairs are

There are 6 pairs, so

(b) Sum greater than 9: The possible sums are 10, 11, and 12. Their numbers of outcomes are 3, 2, and 1 respectively. Hence,

(c) Sum divisible by 3: The possible sums are 3, 6, 9, and 12, with respectively 2, 5, 4, and 1 outcomes. Therefore,

The total number of outcomes is

At least one six: It is easier to use the complementary event, no six. The probability of no six on one die is , so

Therefore,

Exactly two sixes: Select the two dice showing six in

ways. The remaining die must show any of the five non-six values. Thus, the number of favorable outcomes is .

A standard deck contains 52 cards, with 4 suits and 13 cards in each suit.

(a) A king: There are 4 kings.

(b) A red card: There are 26 red cards.

(c) A face card: The face cards are jacks, queens, and kings. There are face cards.

(d) A red king: There are 2 red kings.

There are 4 aces and 48 non-aces in a deck of 52 cards.

Both cards are aces:

One ace and one king: This can occur in two mutually exclusive orders:

• Ace followed by king:

• King followed by ace:

Therefore,

The probabilities differ because the second draw is made without replacement.

For any two events and , the addition theorem is

The intersection is subtracted because it is counted once in and once in .

Let be the event of drawing a heart and the event of drawing a king.

There are 13 hearts, 4 kings, and 1 card that is both a heart and a king, namely the king of hearts.

Therefore,

Hence, the probability is .

The conditional probability of event given that event has occurred is defined by

Rearranging gives the multiplication theorem:

It can also be written as

There are 12 face cards in a standard deck: 4 jacks, 4 queens, and 4 kings. Of these, 4 are kings. Therefore, given that the card is a face card,

Hence, the required probability is .

Let denote the event that the first ball is blue and denote the event that the second ball is green.

Initially, there are 12 balls. If the first ball is known to be blue, then one blue ball has been removed. The box now contains:

• 5 red balls,
• 3 blue balls,
• 3 green balls.

Thus, 11 balls remain, of which 3 are green.

Therefore,

Hence, the probability that the second ball is green given that the first ball is blue is .