Find the value of ₃P₃ and ₃C₃. Explain the difference between these two methods of counting.

A ribonucleic acid (RNA) is a messenger molecule associated with deoxyribonucleic acid (DNA). RNA is made up of a "chain" (sequence) of smaller molecules called nucleotides. The nucleotides contain the bases: U(uracil), C(cytosine), G(guanine), and A(adenine). It is difficult to observe exactly what an entire RNA chain looks like, however it is possible to observe fragments of a chain by breaking it up with certain enzymes. Knowledge about these fragments can sometimes determine the makeup of an entire chain of RNA. The "G-enzyme" will break an RNA chain after each G(guanine) link. The "U-C-enzyme" will break an RNA chain after every U(uracil) and every C(cytosine). Consider the RNA chain AGUGGAUUGUCAUGA. A G-enzyme will break this chain into the fragments AG, UG, G, AUUG, UCAUG, and A. While the U-C-enzyme will break the same chain into the fragments AGU, GGAU, U, GU, C, AU, and GA. Unfortunately, the fragments of a broken-up chain may be mixed up and in the wrong order.
Suppose an RNA chain is broken by a G-enzyme into the fragments AUG, AAC, CG, and AG. While the U-C-enzyme breaks the same RNA chain into the fragments GC, GAAC, and AGAU. What is the complete RNA chain of 10 bases?

Given the 4 letters A, B, C, and D. How many different 5-letter sequences are possible if letters can be repeated?

A deoxyribonucleic acid (DNA) molecule is made up of a "chain" (sequence) of smaller molecules called nucleotides. The nucleotides contain the bases A(adenine), C(cytosine), G(guanine), and T(thymine). In 1952 (building on their predecessors' research in genetics) James Watson and Francis Crick realized that the DNA molecule was too thick to be a single strand. After trying several models, they made one in which two strands were wrapped around each other (a twisted ladder). Today this twisted-two-strand model (called a double helix) is accepted as the correct structure for DNA. In addition, in the early 1950s American scientist Edwin Chargoff made an important discovery about the four nitrogenous bases. Chargoff's work lead to the discovery that across the rungs of the DNA twisted ladder Adenine always pairs with Thymine and Cytosine always pairs with Guanine.
Given this information, how many different five molecule sequences are possible?