Understanding Combination, Slides of Mathematics

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Illustrates the combination of objects; Find the combination of objects taken r at a time; and Value the concepts of combinations in solving real-life situations.

Objective

s

Activity 1:

Recall-ection

How many possible permutations are there in the letters of the word PHILIPPINES? 𝑷 = 𝟏𝟏 × 𝟏𝟎 × 𝟗 × 𝟖 × 𝟕 × 𝟔 × 𝟓 × 𝟒 × 𝟑 × 𝟐 × 𝟏 ( 3 × 2 × 1 )( 3 × 2 × 1 ) Given: (^) 𝒑 =𝟑 2 𝑷 =𝟏 𝟏𝟎𝟖 𝟖𝟎𝟎 Solution: 3 𝑷 = 𝒏! 𝒑! 𝒊!

In how many ways can 10 plates be set on a round table? Given: 3 𝑷 =𝟑𝟔𝟐 𝟖𝟖𝟎 Solution:

Activity 2:

Put Some Order

Here

Answer:

Choosing 5 questions to answer out of 10

questions in a test.

ONI

Order is Not Important

Answer:

Winning in a contest

OI

Order is Important

Answer:

Selecting 7 people to form a Student Affairs

Committee

ONI

Order is Not Important

Answer:

Drawing a set of 6 numbers in a lottery

containing numbers 1 to 45.

ONI

Order is Not Important

Answer:

Entering the PIN (Personal Identification

Number) of your ATM card

OI

Order is Important

Recall that a permutation is an ordered arrangement of objects. Suppose we have a set of 4 letters {A,B,C,D }. In how many ways can we select a 2 letters from this set? Suppose we had to find the number of arrangements of 2 letters possible from those 4 letters. How many arrangement do we have? 𝒏𝑷𝒓 = 𝒏! ( 𝒏 − 𝒓 )! 𝟒 𝑷 𝟐= 𝟒! (𝟒 − 𝟐)! 𝟒 𝑷 𝟐= 𝟒 × 𝟑 × 𝟐 × 𝟏 𝟐 × 𝟏 𝟒 𝑷 𝟐=𝟏𝟐 𝟏𝟐 𝒂𝒓𝒓𝒂𝒏𝒈𝒆𝒎𝒆𝒏𝒕𝒔 In how many ways can the 2 letters be chosen?

Let us say balls 1, 2 and 3 are chosen from pool balls. In how many ways can we arrange them if order does matter?

A combination is the choice of r things from a set of n

things without replacement and where order does not

matter

Formula:

Wherein:

𝒏 = 𝒏𝒐. 𝒐𝒇 𝒐𝒃𝒋𝒆𝒄𝒕𝒔 𝒓 = 𝒏𝒐. 𝒐𝒃𝒋𝒆𝒄𝒕𝒔 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕 𝒕𝒐 𝒔𝒆𝒍𝒆𝒄𝒕

Example 1: A soloist is auditioning for a musical play. If she is required to sing any 3 of the 7 prepared songs, in how many ways can she make her choice? Given: (^) 𝒓 =𝟑 Solution: 𝟕 𝑪 𝟑=

𝟕 × 𝟔 × 𝟓 × 𝟒 × 𝟑 ×𝟐 × 𝟏

𝟑! (𝟒 × 𝟑 × 𝟐 × 𝟏)

𝟕 𝑪 𝟑=𝟑𝟓 Thus, there are 35 ways can she make her choice.