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

Question

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.

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

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