How many ways can $25$ identical pencils be distributed between two people? Each pencil must be given out.
a) Each person must have at least $5$ pencils.
b) Each person must have at least $7$ pencils.
c) Each person must have at least $10$ pencils.
Thank you for your input of wisdom.
Regards!
If each of A and B must have $5$ pencils at least, give them each $5$ pencils. That leaves $15$ pencils. We can give $0$ of these to A, or $1$, or $2$, and so on up to $15$.
The other questions are done in exactly the same way.