A DNA sequence consists of a series of letters representing a DNA strand that spells out the genetic code. There are four possible letters (A, C, G, and T), each representing a specific nucleotide base in the DNA strand (adenine, cytosine, guanine, and thymine, respectively). How many distinguishable sequences can be formed using two As, two Cs, three Gs, and one T?

Approach Each sequence formed will have eight letters. To construct each sequence, we need to fill in eight positions with the eight letters:

The process of forming a sequence consists of four tasks:
Task 1: Choose the positions for the two As.
Task 2: Choose the positions for the two Cs.
Task 3: Choose the positions for the three Gs.
Task 4: Choose the position for the one T.

Task 1 can be done in ₈C₂ ways because we are choosing the 2 positions for A, but order does not matter (because we cannot distinguish the two As). This leaves 6 positions to be filled, so task 2 can be done in ₆C₂ ways. This leaves 4 positions to be filled, so task 3 can be done in ₄C₃ ways. The last position can be filled in ₁C₁ way.

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