How many routes can be formed go 3 blocks west and 4 blocks north, assuming you can only move in these directions?

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BlackTom AI · Accepted Answer

The question asks how many routes exist to go 3 blocks west and 4 blocks north when only moves west or north are allowed. The total number of moves to reach the destination is 3 + 4 = 7, consisting of exactly 3 west moves (W) and 4 north moves (N).

Option a: 18. This would require counting sequences incorrectly, perhaps by misallocating positions for W or N. However, the correct count is a binomial arrangement of 7 moves with 3 identical W moves, which yields 7 choose 3 = 35, not 18.

Option b: 24. This value does not match the combinatorial count for choosing which of the 7 positions are the 3 west moves. If you try to construct a sequence, you would still end up distributing 3 W among 7 slots, which gives 35 possibilities, not 24.

Option c: 12. This is significantly off from the true combinatorial count. It might come from assuming fewer total sequences or miscounting, but with 7 total moves and 3 of them being West, the number of unique orderings cannot be as low as 12.

Option d: 35. The proper way to count is to choose which 3 of the 7 total moves are the westward steps (or which 4 are the northward steps). This is a classic combinations problem: number of sequences = C(7,3) = 7! / (3! 4!) = 35. Thus the value 35 aligns with the correct combinatorial reasoning and edge constraints (only W and N moves, exactly 3 W and 4 N).

In summary, the correct reasoning hinges on treating the route as a 7-step sequence with exactly 3 W and 4 N, and counting the distinct permutations of these moves. The other options diverge from this combinatorial count and thus are not equal to the true number of routes.

MDM4U - Mathematics of Data Management 12 (2025-26) - A

How many routes can be formed go 3 blocks west and 4 blocks north, assuming you can only move in these directions?

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