Exactly four different toy cars will be chosen to serve as prizes at a child's birthday party. There are seven toy cars available: a yellow coupe, a green sedan, a red convertible, a black all-terrain vehicle, a gold utility vehicle, a silver hatchback, and a red pickup truck. Each toy car selected for the party must be a different color. If the green sedan is chosen for the party, then the red convertible must also be chosen. Exactly two of the following three cars must be chosen: the yellow coupe, green sedan, and red convertible. Select a toy car that must be chosen for the party. Then select a toy car that cannot be chosen for the party. Make only two selections, one in each column.

I have a doubt here here’s my analysis: Condition 1: Each toy car selected for the party must be a different color Condition 2: Exactly two of the following three cars must be chosen: the yellow coupe, green sedan, and red convertible. Also we have 2 red color vehicle and other all different coloured. But as per condition 1, we know that we cannot have two same coloured vehicles so one red vehicle cannot be chosen. As per condition 2, we can select exactly two of the three: yellow, green or red! Here I have choice to select any two. if I don’t choose red convertible still I can eliminate red pick up truck and all the vehicles will have different color We can safely eliminate red pick up truck. But why is red convertible “must be chosen”? Thanks Harsh Tandel Posted from my mobile device

Harsh basically answered everything. I just wanted to solve his doubt. As per condition 2, we can deduce 2 cases- 1. Yellow and red convertible 2. Green and red convertible You have to pick from either of case as it is compulsory to pick 2 from the three cars. Red convertible is fixed in both choices and since you can't repeat colours also. The red convertible is a "must be chosen".

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Exactly 4 different cars are selected.
All selected cars must have different colors → since RC and RPT are both red, at most one of them can be chosen.
If GS is selected, then RC must also be selected (GS → RC).
Exactly two of {YC, GS, RC} must be selected.
Step 3: Link the Rules
Look at all possible ways to pick exactly two from {YC, GS, RC} and check rule 3:
{YC, GS}
GS is selected ⇒ RC must also be selected.
That would give {YC, GS, RC} = three cars from this set, which violates “exactly two”.
So {YC, GS} is impossible.
{YC, RC}
GS is not selected, so rule 3 doesn’t apply.
This combination is allowed.
{GS, RC}
GS is selected and RC is selected, satisfying rule 3.
This combination is also allowed.
In both valid cases ({YC, RC} and {GS, RC}), RC is always selected.
👉 Therefore, Red Convertible (RC) must be selected.
Now combine this with the color rule (all colors must be different):
RC is red, and it must be selected.
RPT is also red.
So RPT cannot be selected (otherwise we’d have two red cars).
Step 4: Quick Check of Other Cars
YC and GS: sometimes selected, sometimes not → not “must”.
BATV, GUV, SH: can be chosen or not, depending on how we fill up to 4 cars → not “must” and not “cannot”.
RC: appears in all valid combinations → must be selected.
RPT: conflicts with RC’s color → cannot be selected.
Final Answer
Car that must be selected: Red Convertible (RC)
Car that cannot be selected: Red Pickup Truck (RPT)

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