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?

286

Step-by-step explanation:

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

To write a mixed fraction as an improper fraction, simply multiply the denominator by the whole number and add on the result to the numerator. The denominator part will remain the same throughout. 



In this case:
- 10 is the whole number component
- 3 is the numerator
- 4 is the denominator

Multiple the denominator (4) by the whole number component (10). 
4×10=40

Add the numerator (3) to the answer you had in the previous step (40).  
3+40=43

Since the denominator remains same throughout, in this case,  as an improper fraction would be