Problem Solving

From a class of 25 students, 10 are to be chosen for an excursion party. There
are 3 students who decide that either all of them will join or none of them will
join. In how many ways can the excursion party be chosen ?

I have adoubt in approach here, I could think of two ways to solve this:-

[spoiler]1) consider the three students as one entity
thus total students now are 22+1=23
ways of selecting 10 out of 23 = 23C10

2) there are 22 + 3 students
ways of selecting 7 from 22( these 3 are there in the selected lot) = 22C7
ways of selecting 10 from 22 (these 3 are not selected) = 22C10
total ways 22C7 + 22C10

but 23C10 is not equal to 22C7 + 22C10 [/spoiler]

[spoiler]that means there is a mistake in either of the approach, but where? I can't figure it out!![/spoiler]

In your first scenario, you're saying you have 22 students plus one group of 3 students who can't be separated for a total of 23. But if you do 23C10, then if the supergroup is one of the ten selected, you would actually have 12 total students instead of 10.