can someone please help me solve this? Mark and Eric play in a tennis tournament

Question

can someone please help me solve this? Mark and Eric play in a tennis tournament. The first person to win two games in a row or three wins the tournament. Draw a tree diagram to show the possible outcomes of the tournament. (a) How many four digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5 if no digit may be repeated? (4 digit number cannot have a leading zero.) (b) How many four digit numbers formed in part (a) are even? (c) How many are divisible by five? (d) How many are less than 3000?

Explanation / Answer

Solution:-

6). Given digits - 0,1,2,3,4,5

a). Four digit number that can be formed = 5 * 5 * 4 * 3

That is for the first digit we can choose any of the 5 digits (excluding 0), for second digit we can choose any of the remaining 5 digits(including zero) and so on.

5 * 5 * 4 * 3 = 300

b). Four digit number that are even

That is the number can end with 0, 2 or 4

First digit cannot be zero, so the number of choices only 5 (1, 2, 3, 4, 5)

The last digit can be pick from 0, 2, 4 so the number of choices only 3

Second digit can be only pick from the rest, so the number of choices only 4

Third digit can be only pick from the rest, so the number of choices only 3

The total number of choices is 5*3*4*3 = 180

c). Divisible by 5

That is the number can end with a 0 or 5.

First digit cannot be zero, so the number of choices only 5 (1, 2, 3, 4, 5)

The last digit can be 0 or 5 so the number of choices only 2

Second digit can be only pick from the rest, so the number of choices only 4

Third digit can be only pick from the rest, so the number of choices only 3

The total number of choices is 5*2*4*3 = 120

d). Less than 3000.

That is the first digit can be 1 or 2.

Largest number that can be formed is 2543 and the smallest that can be formed is 1023

First digit cannot be 0 or 3 or 4 or 5, so the number of choices only 2 (i.e., either 1 or 2)

The last digit can be pick from the rest, so the number of choices is 5

Second digit can be only pick from the rest, so the number of choices only 4

Third digit can be only pick from the rest, so the number of choices only 3

The total number of choices is 2*5*4*3 = 120

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