Calculas Scholarship T1W11

--- title: Calculas Scholarship T1W11 tags: Calculus Scholarship description: Term 1 Week 11 type: pdf slideOptions: theme: white --- ###### tags: `Calculus Scholarship` ### 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 $7 \choose 3$ $= \frac{7!}{3!4!} =35$ ### 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. ### Q3 ### Any 3 lines would form a triangle. So just choose 3 from 7 lines. Which is $7 \choose 3$ $= 35$ triangles ### 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. ### 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