Ali decides to make some pretty patterns using 12 indistinguishable balls. He creates a 12 × 12 grid to place the balls cones on. How many patterns can he make such that no two balls are in the same row or in the same column of the grid?
If the balls are distinguishable, how many ways are there for Ali to place the balls on the 12 × 12 grid now, such that no two balls are in the same row or in the same column of the grid?
Say that there are now 2 white, 3 brown, and 7 green balls. Balls of the same type are indistinguishable, but balls of different types are distinguishable. How many ways can Taki arrange the balls now such that no two balls are in the same row or in the same column of the 12 × 12 grid?
Miranda is arranging 30 distinguishable necklaces for display in her jewelry store such that there are two rows of 15 necklaces each. Of the 30 necklaces , there are 7 special distinguishable diamond necklaces and 3 special distinguishable emerald necklaces. She decides that the 7 diamond necklaces must be placed in the front row, and the 3 emerald necklaces must be placed in the back row. How many ways can the 30 necklaces be arranged?
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