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On the grid shown, moves are only right or up from A to B. How many shortest routes pass through P but avoid the closed crossing X?

Competition practicePermutations & Combinations

A30
B18
C70
D12

Step-by-Step Solution

Through P(1, 2):

^{3} C_{1} \times^{5} C_{3} = 3 \times 10 = 30

. Through P and X(3, 3):

3 \times^{3} C_{1} \times^{2} C_{1} = 3 \times 3 \times 2 = 18

. Through P but not X:

30 - 18 = 12

.

Answer: D) 12

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