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The number of standard Young tableaux of shape λ is the number of labelings of grid cells in the Young diagram of λ with distinct integers 1,...,|λ|, times the probability that a random such labeling is a standard Young tableau. The number of such labelings is |λ|!.
For every grid cell, the probability that its label is less than everything to its right or below it is the reciprocal of its hook length (the number of grid cells to its right, plus the number of grid cells below it, plus one for itself).
If these events were independent, then the probability of a random labeling of the Young diagram being a standard Young tableau would be the product of reciprocals of all the hook lengths.
And hence the number of standard Young tableaux of shape λ would be |λ|! divided by the product of hook lengths. These events are not independent. This formula holds anyway.
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