Answer:
a. 390 ways
b. 0.82
Explanation
Given
Number of guard = 2
Forward = 2
Center = 1
a.
The college team has on its roster 3 centers, 4 guards, 5 forwards, and one individual (X)
Considering the lineups without X, then lineups with X as guard, then lineups with X as forward.
Note that order doesn't matter
Calculating the lineup without player X:
Selection of 1 Center from 3 center = 3C1 = 3
Selection of 2 forwards from 5 forwards = 5C2 = 10
And
Selection of 2 guards from 4 = 4C2 = 6
Total = 6 * 10 * 3 = 180
X as a guard: (only one player is needed since X has already been selected)
Selection of 1 Center from 3 center = 3C1 = 3
Selection of 2 forwards from 5 forwards = 5C2 = 10
And
Selection of 1 guards from 4 = 4C1= 4
Total = 4 * 10 * 3 = 120
X as a forward: (only one player is needed since X has already been selected)
Selection of 1 Center from 3 center = 3C1 = 3
Selection of 1 forwards from 5 forwards = 5C1 = 5
And
Selection of 2 guards from 4 = 4C2= 6
Total = 6 * 5 * 3 = 90
Total Selection = 90 + 120 + 180 = 390 ways
b.
Considering that player X and Y sit out, player X play while Y sits out, player Y plays while X sits out and player X and Y play (either as a guard, forward or split duties).
Calculating the lineup without player X and Y
Selection of 1 Center from 4 center = 4C1 = 4
Selection of 2 forwards from 4 forwards = 4C2 = 6
And
Selection of 2 guards from 4 = 4C2 = 6
Total = 4 * 6 * 6 = 144
Calculating the lineup without player X but not Y
X as a forward: (only one player is needed since X has already been selected)
If X plays as forward
Selection of 1 Center from 4 center = 4C1 = 4
X plays as forward = 4C1 = 4
And
Selection of 2 guards from 4 = 4C2= 6
Total = 4 * 4 * 6 = 96
If X plays as guard
Selection of 1 Center from 4 = 4C1 = 4
Selection of 2 forwards from 4 = 4C2 = 6
And
X as guard = 4C1= 4
Total = 4 * 6 * 4 = 96
So, if X play and Y sits out; total = 96 + 96 = 192
It's same if Y plays and X sits out = 192
Considering that both X and Y plays
As guards
Selection of 1 Center from 4 = 4C1 = 4
Selection of 2 forward from 4 = 4C2 = 6
And
X and Y as guard = 4C0 = 1
Total = 4 * 6 * 1 = 24
As forwards
Selection of 1 Center from 4 = 4C1 = 4
Selection of 2 guards from 4 = 4C2 = 6
And
X and Y as forward = 4C0 = 1
Total = 4 * 6 * 1 = 24
Both on split duties
Selection of 1 Center from 4 = 4C1 = 4
Selection of1 from 4 forward= 4C1 = 4
And
One from 4 forward = 4C1= 4
Total = 4 * 4 * 4 = 72
Total = 72 +24 + 24 = 120
Total Required Selection = 144+192+192+120 = 648
Overall Possible Selection = 12C5 = 792
Probability = 648/792 = 0.82 ----- Approximated
