PLEASE WRITE NEATLY EXPLAIN YOUR ANSWER IF POSSIBLE SO THAT

PLEASE WRITE NEATLY, EXPLAIN YOUR ANSWER IF POSSIBLE SO THAT I MAY UNDERSTAND. PLEASE USE LEAST COMPLICATED SOLUTIONS. THANK YOU.

1) If there was a group of four students who were all asked to select an integer between 7 and 27.

a) How many permutations are possible for the numbers selected?

b) How many permutations are possible with four different values?

c) How many permutations are possible with three different values and one duplicate?

d) How many permutations are possible with all four values the same?

e) How many permutations are possible with exactly three values the same?

f) How many permutations are possible with exactly two pairs of identical values the same?

g) What is the probability that all four are the same?

h) What is the probability that all four are different?

i) What is the probability that exactly 3 value will be the same?

2) Assume that we have a hexahedral die with the six values 1, 3, 5, 7, 8 and 10 on the die that is rolled 3 times.

a) What is the probability that all 3 values will be the same?

b) What is the probability that all 3 values will be the different?

c) What is the probability that all exactly 2 of the values will be the same?

Solution: given, no. between 7 & 27 is 19, 4 student

a]: 4 salected from 19 is ¹⁹C₄ =3876

b]: possible permutation of diffrent no. is=¹⁹P₄=93024

c]: possible permutation=19*18*17*3=17442 ( in this case we consider duplicate means one no. salected from already slected no.)

d]: possible permutation for same value is=19*19*19*19=130321

e]: possible permutation exact three value the same=19*19*19*18=123462

f]: possible permutation for two pair of exactly the same=19*19*2

g]: probability for all the same= 1/19*1/19*1/19*1/19

h]: probability for all the diffrent= 1/19*1/18*1/17*1/16

i]: probability for three the same= 1/19*1/19*1/19*1/18

2]: a]: probability that all 3 values will be the same =1/6*1/6*1/6

b]: probability that all 3 values will be the diffrent=1/6*1/5*1/4

c]: probability that all 2 values will be the same =1/6*1/6*1/5