Final answer:
The total number of 20-digit quaternary sequences with at least one 2 and an odd number of 0's is 73,619,579,470.
Step-by-step explanation:
To find the number of 20-digit quaternary sequences where there is at least one 2 and an odd number of 0's, we can break down the problem into two cases:
Case 1: There is exactly one 2 and an odd number of 0's.
In this case, we can choose the positions for the 2 and the odd number of 0's. There are 20 choices for the position of the 2, and for each of the remaining 19 positions, we can choose whether it is a 0 or one of the other three quaternary digits (1, 3). Thus, we have:
20 (choices for the position of 2) * 219 (choices for the remaining digits) = 20 * 524,288 = 10,485,760 sequences.
Case 2: There are two or more 2's and an odd number of 0's.
In this case, we can choose the positions for the 2's and the odd number of 0's. There are a total of 20 choose 2 = 190 combinations for the positions of the 2's. For each combination, we can choose the remaining digits in the remaining 18 positions from the three remaining quaternary digits (0, 1, 3). Thus, we have:
190 (choices for the positions of 2's) * 318 (choices for the remaining digits) = 190 * 387,420,489 = 73,609,093,710 sequences.
Therefore, the total number of 20-digit quaternary sequences where there is at least one 2 and an odd number of 0's is:
10,485,760 (from Case 1) + 73,609,093,710 (from Case 2) = 73,619,579,470 sequences.