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Combinatorics, EXT1 A1 2022 HSC 7 MC

The diagram shows triangle `A B C` with points chosen on each of the sides. On side `A B`, 3 points are chosen. On side `A C`, 4 points are chosen. On side `B C`, 5 points are chosen.
 

How many triangles can be formed using the chosen points as vertices?

  1. 60
  2. 145
  3. 205
  4. 220
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`C`

Show Worked Solution

`text{1 point taken from each side:}`

`text{Triangles} = 3 xx 4xx5=60`
 

`text{2 points taken from one side:}`

`text{Triangles}=((3),(2))((9),(1))+((4),(2))((8),(1))+((5),(2))((7),(1))=145`

`:.\ text{Total triangles} =60+145=205`

`=>C`


♦♦ Mean mark 37%.

Combinatorics, EXT1 A1 EQ-Bank 27

Eight points `P_1, P_2, ..., P_8`, are arranged in order around a circle, as shown below.
 

  1. How many triangles can be drawn using these points as vertices?   (1 mark)

  2. How many pairs of triangles can be drawn, where the vertices of each triangle are distinct points?   (2 marks)

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a.    `56`

b.    `280`

Show Worked Solution

a.    `text{Total triangles}= \ ^8 C_3= 56`

COMMENT: In part (ii), divide by 2 to account for duplicate pairs.

 
b.    `text{Total pairs}= (\ ^8 C_3 xx \ ^5 C_3)/{2}= 280`

Combinatorics, EXT1 A1 EQ-Bank 30

How many rectangles, including all squares, can be found in the 4 × 5 grid below, in total?   (2 marks)
 

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`150`

Show Worked Solution

`text{Consider the 5 horizontal lines}`

`text{Height combinations of rectangle} =\ ^5C_2`

`text{Consider the 6 vertical lines}`

`text{Width combinations of rectangle} =\ ^6C_2`

`:. \ text{Total rectangles in grid} = \ ^5C_2 xx \ ^6C_2 = 150`

Combinatorics, EXT1 A1 2017 HSC 10 MC

Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square.
 

What is the probability that all three crosses lie in the same row, column or diagonal?

  1. `1/28`
  2. `2/21`
  3. `1/3`
  4. `8/9`
Show Answers Only

`B`

Show Worked Solution

`P= text(favourable events)/text(total possible events)= (3 + 3 + 2)/(\ ^9C_3)= 2/21`

`=>B`