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CAT 2019 Slot 1 — QA Question 2

Geometry based questionsEasy

With rectangular axes of coordinates, the number of paths from (1, 1) to (8, 10) via (4, 6), where each step from any point (x, y) is either to (x, y + 1) or to (x + 1, y), is

Answer & solution

Correct answer: 3920

Answer: 3920

Solution

Easy

Each step goes right or up. A path forced through (4,6)(4,6) is just a path to (4,6)(4,6) followed by a path from (4,6)(4,6) to (8,10)(8,10), so multiply the two counts. For a grid trip needing hh right-steps and vv up-steps, the number of paths is (h+vh)\binom{h+v}{h}.

1

Leg 1: (1,1)→(4,6)(1,1)\to(4,6). Need 4−1=34-1=3 right-steps and 6−1=56-1=5 up-steps, total 88 steps.

N1=(83)=56(choose positions of the 3 right-steps)\begin{aligned} &N_1 = \binom{8}{3} = 56 \quad\text{(choose positions of the 3 right-steps)} \end{aligned}
2

Leg 2: (4,6)→(8,10)(4,6)\to(8,10). Need 8−4=48-4=4 right-steps and 10−6=410-6=4 up-steps, total 88 steps.

N2=(84)=70(choose positions of the 4 right-steps)\begin{aligned} &N_2 = \binom{8}{4} = 70 \quad\text{(choose positions of the 4 right-steps)} \end{aligned}
3

Multiply the legs. Every leg-1 path combines with every leg-2 path.

N=N1⋅N2=56×70=3920[(step 1)(step 2)]\begin{aligned} &N = N_1\cdot N_2 = 56\times 70 = 3920 \quad\text{[(step 1)(step 2)]} \end{aligned}
N=3920N = 3920

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CAT 2019 Slot 1 QA Q2: With rectangular axes of coordinates, the number of paths from (1, 1) to (8, 10) via (4, 6), where each step f — Solution | TheCATExam