1.  If you can use a dollar coin, then you have to make 
the remaining 35 cents with 9 coins, which is 5 pennies, 2 dimes
and 2 nickels.   That's 1 way, and the only way using a dollar
coin.

2.  If you don't use a dollar coin but can use 2 half dollars, 
then you have to make the remaining 35 cents with 8 coins, which is 
5 pennies and 3 dimes. That's 1 way, and the only way using 2 half dollars.

3.  If you don't use any dollar coins but can use 1 half dollar, then
you have to make the remaining 85 cents with 9 coins.  There is one way 
each of making 85 cents using 0,1,2, or 3 quarters with 9,8,7,or 6 coins,
respectively:

    a.  If you don't use any quarters, you have to use 8 dimes and 1 nickel
        to make the remaining 85 cents with 9 coins.
    b.  If you use 1 quarter, you have to use 4 dimes and 4 nickels
        to make the remaining 60 cents with 8 coins.
    c.  If you use 2 quarters, you have to use 7 nickels
        to make the remaining 35 cents with 7 coins.
    d.  If you use 3 quarters, you have to use 1 nickel and 5 pennies
        to make the remaining 10 cents with 6 coins.

That's 4 more ways.

4.  If you don't any dollar coins or half dollars, then you can use
    3 or 4 quarters.

    a.  If you use 3 quarters, you have to use 5 dimes and 2 nickels
        to make the remaining 60 cents with 7 coins.
    b.  If you use 4 quarters, you have to use 1 dimes and 5 nickels
        to make the remaining 35 cents with 6 coins.

That's 2 more ways.

So that's a grand total of 1+1+4+2 = 8 ways to make $1.35 with 10 coins.
If your teacher doesn't want you to use dollar coins or half dollars, then
you should eliminate those from the 8 ways.

Edwin