Let \(A = \{1, 5, 9, 11, 15, 23\}\)
- Find the number of sequences of length 3 using elements of A.
Since elements can repeat we have: \(6^3 = 216\)
- Repeat part (a) if no element of A is to be used twice.
\(P(6,3) = 6 \cdot 5 \cdot 4 = 120\)
- Repeat part (a) if the first element of the sequence is 5.
Since \(5\) is fixed in the first position the two remaining positions can be filled with any of the six elements: \(6^2 = 36\)
- Repeat part (a) if the first element of the sequence is 5 and no element of A is used twice.
\(P(5,2) = 5 \cdot 4 = 20\)