---
# System prepended metadata

title: Calculas Scholarship T1W11
tags: [Calculus Scholarship]

---

---

title: Calculas Scholarship T1W11 
tags: Calculus Scholarship 
description: Term 1 Week 11 
type: pdf
slideOptions:
 theme: white

---

###### tags: `Calculus Scholarship`

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### Q1 ###

There are 7 steps from A to B. So we pick 3 of 7 to be down steps, then the rest 4 would be right steps. Which is $7 \choose 3$ $= \frac{7!}{3!4!} =35$

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### Q2 ###

The Paths can be categorised into two groups, go through the Blue point, then to B, or go through the Purple point, then to B.

For the first group, from A to Blue, there are $5 \choose 2$ paths, from Blue to B, there are $5 \choose 2$ paths again, so overall $10 \times 10 = 100$ paths;

For the second group, there are also 100 paths just like above, so in total 200 paths.

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### Q3 ###

Any 3 lines would form a triangle. So just choose 3 from 7 lines. Which is $7 \choose 3$ $= 35$ triangles

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### Q4 ###

There are 8 lines, so $8 \choose 3$ = $56$ chooses, in which there are two situations that cannot form a triangle, the first situation is to $5 \choose 3$ $=10$ from the positive slope lines, another situation is $4 \choose 3$ $=4$ from the negative slope lines. The other situation all form triangles, then it is $56 - 10 - 4 =42$ triangles.


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### Q5 ###

a). Permutation of 7 objects, $7! = 5040$ ways

b). Permutation of 6 objects, times the permutation of 2 objects, which is $6! \times 2! = 720 \times 2 = 1440$ ways

c). Two way of thinking:

- 1st, 5040 - 1440 = 3600 ways
- 2nd, Permutation of 5 boys times choosing two from 6 gaps, then times permutation of 2 girls, which is $5! \times$ $6 \choose 2$ $\times 2=120\times 15 \times 2 = 3600$ ways

d). $5 \choose 1$ $\times 5! \times 2! = 5 \times 120 \times 2 =1200$ ways

e). $5 \choose 2$ $\times 4! \times 2! \times 2!= 10 \times 24 \times 4 = 960$ ways

f). $5 \choose 3$ $\times 3! \times 2! \times 3!= 10 \times 6 \times 12 = 720$ ways

g). $5 \choose 4$ $\times 2! \times 2! \times 4!= 5 \times 4 \times 24 = 480$ ways

h). $5! \times 2! = 120 \times 2 = 240$ ways

