please help me ( probabilities )

Question

please let me know if my answer true or false Three numbers are chosen at random without replacement from the set {0, 1, 2, 3, ... , 10}. Calculate the probabilities that for the three numbers drawn

i) None will be greater than 7,

8*7*6 / 11*10*9 = 56 / 165

ii) the smallest will be 7,

4*3*2 / 11*10*9 = 4 / 165

iii) the sum is equal to 7.

4*3*2 / 11*10*9 = 4 / 165

Answer 1

iii)

The laplace definition of probability is the ratio of the number of favored events to the number of total possible events

number of total possible events:$11\cdot 10\cdot 9$

For the number of favored Events I would write down all the combinations with the sum of 7:

016 (6)

025 (6)

034 (6)

124 (6)

To get all the permutations you have to sum all the number in the brackets.

For example 016 can be written as: 016. 061, 160, 106, 601, 610

greetings,

calculus

Answer 2

Here $\binom{11}{3}=165$ serves as denominator.

i) is okay.

ii) there are $3$ possibilities (leading to probability $\frac{3}{165}$):

$7-8-9$

$7-8-10$

$7-9-10$

iii) $\frac{4}{165}$ is the correct answer and can be deduced from the fact that there are $4$ possibilities (wich are all mentioned). However, I cannot find any connection with the factorization $\frac{4}{11}\times\frac{3}{10}\times\frac{2}{9}$.