Count the number of ways with no A's and subtract from the total number of sequences.3. How many 8-letter sequences are there consisting of As, Bs, and Cs with the following restrictions?

a)The sequence must have at least one A.

There are sequences with A,B,C. There are ways with

B and C.

Count the sequences that have NONE of each symbol and subtract . By that, I assume they mean it must contain one of each. So, all of one kind or two different ones are allowed. You could have all A's, all B's, all C's, B and C, A and B, A and Cb)The sequence must have at least one of each symbol.

4. A necklace is made out of identically shaped, coloured beads. Ten beads are red, 9 are black, 7 are white, 12 are pink, and 4 are purple. The beads are strung together on a waxed string. The ends are tied together to form a large knot that prevents the beads from sliding past it. How many different necklaces can be made in this way?

There are 42 beads. So the arrangements are

We divide by two to keep from overcounting because the necklace can be turned around.

I hope I didn't miss something.