20. (3 points) As shown in the figure, the electronic clock can display times from 00:00:00 to 23:59:59. In one day (24 hours), the number of times the clock displays a time exactly composed of the digits $1,2,3,4,5,6$ is $\qquad$ kinds.
【Analysis】Since the first digit of the hour can only be 1 or 2, the first digit of the minute and second can be at most 6, and when 6 is not on the hour, 6 can only be in the second position of the minute or second. According to this, we will discuss the cases. Based on this, we solve the problem.

【Solution】Solution: First, look at the hour: it can only be (1) 12, (2) 13, (3) 14, (4) 15, (5) 16, (5 cases starting with 1) (6) 21, (7) 23 (2 cases starting with 2). Next, look at the minute and second: when 6 is not displayed on the hour, 6 must be in the unit place! (1) When the hour is 12, the minute and second are as follows: 6 in the second position of the minute, the other three digits can be in any position, there are $3645,3654,4635,4653,5634,5643$ a total of 6 cases, 6 in the second position of the second, the other three digits can be in any position, there are $3456,3546,4356,4536,5346,5436$ a total of 6 cases, a total of 12 cases. (2) When the hour is $13,14,15,21,22$, the situation is similar to when the hour is 12, a total of $12 \times 5=60$ cases, (3) When the hour is 16, the minute and second: the digits in each position have no restrictions, they can be in any position: $2345,2354,2435,2453,2534,2543$, (6 cases starting with 2) $3245,3254,3425$, $3452,3524,3542$, (6 cases starting with 3) $4235,4253,4325,4352,4523,4532$, (6 cases starting with 4) $5234,5243,5324,5342,5423,5432$, (6 cases starting with 5) a total of $6 \times 4=24$ cases. In summary, the number of times in a day when the clock displays a time composed of the digits $1,2,3,4,5,6$ is $12+60+24=96$ (cases). Therefore, the answer is: 96.