Let's use this notation for each possibility PXY and NXY, where
the first letter tells whether it's a penny or a nickel. The X will be
either 1, 2, or 3 to indicate which of the the three pennies or which of the
three nickels it is. The Y will represent either heads H or tails T.
For example:
P2H means penny #2 comes up heads, that is X=2 and Y=H
and
N3T means nickel #3 comes up tails, that is, X=3 and Y=T
First we calculate the numerator of the probability.
So we count the number of successful ways we can have
2 pennies and 1 nickel, then 1 penny and 2 nickels, and
then 3 nickels and no pennies.
1. 2 pennies and 1 nickel
PXY PXY NXY
We can choose the X's for the pennies in 3C2 or 3 ways
We can then assign one of the pennies Y=H or Y=T in 2 ways
We can then assign the other penny Y=H or Y=T in 2 ways
We can then choose the X for the nickel in 3 ways
We can then choose the Y for the nickel in only 1 way (heads) Y=H
That's 3*2*2*3*1 = 36 ways
2. 1 penny and 2 nickels.
PXY NXY NXY
We can choose the X for the penny in 3 ways
We can then assign the penny Y=H or Y=T in 2 ways
We can choose X's for the 2 nickels in 3C2 or 3 ways
We can choose the Y's for the nickels in 3 ways, that is, give one of them
T and the other H in 2 ways, and give them both H's which is 1 more way.
That's 3 ways.
That's 3*2*3*3 = 54
3. 3 nickels and no pennies.
N1Y N2Y N3Y
We'll first figure the number of ways to make the Y's anything and then
subtract the 1 case N1T N2T N3T of all tails.
There are 2 ways to choose the Y for N1.
Then there are 2 ways to choose Y for N2.
Then there are 2 ways to choose Y for N3.
That's 2*2*2 or 8 ways minus the 1 way in which all 3 are tails.
So that's 7 ways to have 3 nickels
So the total number of successful ways is 36+54+7=97
--------------
Now we calculate the denominator of the probability.
So we count the number of possible ways we can have
3 pennies no nickels, then 2 pennies and 1 nickel, then 1 penny
and 2 nickels, and 3 nickels and no pennies.
1. 3 pennies and no nickels.
P1Y P2Y P3Y
We can assign the first penny Y=H or Y=T, that's 2 ways.
We can then assign the second penny Y=H or Y=T, that's 2 ways
We can then assign the third penny Y=H or Y=T, that's 2 ways.
So the number of ways we can have 3 pennies is 2*2*2=2³=8 ways.
2. 2 pennies and 1 nickel
PXY PXY NXY
We can choose the X's for the pennies in 3C2 or 3 ways
We can then assign one of the pennies Y=H or Y=T in 2 ways
We can then assign the other penny Y=H or Y=T in 2 ways
We can then choose the X for the nickel in 3 ways
We can then choose the Y for the nickel Y=H or Y=T in 2 ways
That's 3*2*2*3*2 = 72 ways
3. 1 penny and 2 nickels. This is esactly like the previous
case, just swapping nickes and pennies. But I'll go through
it anyway, but it'll come out the same, 72.
PXY NXY NXY
We can choose the X for the penny in 3 ways
We can then assign the penny Y=H or Y=T in 2 ways
We can choose X's for the 2 nickels in 3C2 or 3 ways
We can then assign one of the nickels Y=H or Y=T in 2 ways
We can then assign the other nickel Y=H or Y=T in 2 ways
That's 3*2*3*2*2 = 72
4. 3 nickels. This is the same as for when there are 3 pennies above,
but I'll go through it anyway:
N1T N2T N3T
We can assign the first nickel Y=H or Y=T, that's 2 ways.
We can then assign the second nickel Y=H or Y=T, that's 2 ways
We can then assign the third nickel Y=H or Y=T, that's 2 ways.
So the number of ways we can have 3 nickels is 2*2*2=2³=8 ways.
So the total number of possible ways is 8+72+72+8 = 160
So the desired probability is 97/160
Edwin
