11. A child has a repertoire of four piano pieces
P1, P2, P3, P4. Her parents expect her to practice 10 pieces
each day. For example, one practice session might consist
of P3, played 7 times and P1 played 3 times. How many
possible practice sessions are there?

First, we can tell that this is either a combination or permutation because we have to determine the number of possible combinations based on a limited number of options and a set number of pieces to practice each day.

Next, because order doesn't matter (it doesn't matter if she practices P1 or P3 first, for example), this is a combination.

After that, we can repeat a piano piece, so there is repetition

If there are n things to choose from, and we have to choose r things, our formula for a combination with repetition is

(r+n-1)!/(r!(n-1)!)

Here, we choose from 4 pieces and we need to choose 10 each day, so we have

(r+n-1)!/(r!(n-1)!) = (4+10-1)!/(10!(4-1)!) = 13!/(10!3!) = 13*12*11/(3*2*1) = 286 possible combinations

11. A child has a repertoire of four piano pieces P1, P2, P3, P4. Her parents expect her to practice 10 pieces each day. For example, one practice session might consist of P3, played 7 times and P1 played 3 times. How many possible practice sessions are there?

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11. A child has a repertoire of four piano pieces
P1, P2, P3, P4. Her parents expect her to practice 10 pieces
each day. For example, one practice session might consist
of P3, played 7 times and P1 played 3 times. How many
possible practice sessions are there?