I previously asked this question:
It's asking whether or not killstreak rewards which are given when you open a care package killstreak contribute to challenges.
Do they contribute in Modern Warfare 3 as well?
Yes, they do. However, I believe it is only towards the "Prestige" challenges, and not the ones under the Assault Strike Package challenge menu.
Even though I never take the Assault strike package, I have several levels of multiple killstreak rewards "Prestige" challenges completed because of care packages I picked up that belonged to other players.