smc-1082-50-Geometry

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?

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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 SM-Bank 21

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

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

--- 1 WORK AREA LINES (style=lined) ---
How many pairs of triangles can be drawn, where the vertices of each triangle are distinct points? (2 marks)

--- 2 WORK AREA LINES (style=lined) ---

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`56`
`280`

Show Worked Solution

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

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

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

Combinatorics, EXT1 A1 SM-Bank 20

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

--- 6 WORK AREA LINES (style=lined) ---

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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?

A. `1/28`

B. `2/21`

C. `1/3`

D. `8/9`

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`B`

Show Worked Solution

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

`=>B`