Hard The letters of the word EQUATION are rearranged so that all the vowels appear together as a single block and the first letter of the new word is a consonant. How many such rearrangements are possible?

MCQ From the 26 letters of the English alphabet, in how many ways can a 5 - letter string be formed such that all five letters are distinct, at least two of them are vowels (vowels being A, E, I, O, U), and the letters appear in strict alphabetical order from left to right?

MCQ The letters of the word PROBABILITY are arranged in a random order, with all arrange- ments equally likely. What is the probability that the two I’s have exactly one letter between them in the arrangement?

All 11 letters of the word PROBABILITY are arranged in a row in random order, each ar- rangement equally likely. Find the probability that both occurrences of the letter I appear before both occurrences of the letter B (reading left to right).

A fair six - sided die is rolled three times. Given that the sum of the three rolls is even, find the probability that all three rolls show an even number.

A box contains 5 white and 7 black balls. Three balls are drawn one by one, without replace- ment. Find the probability that the second ball drawn is white, given that the third ball drawn is black.

Three fair six - sided dice are rolled. Given that the product of the three numbers shown is even, find the probability that at least one of the dice shows a 6.

A and B play a game in which they alternately roll a fair six - sided die, with A rolling first. A wins immediately if she rolls a 6. B wins immediately if he rolls a 5 or 6. The game continues until someone wins. Find the probability that A wins the game.

A committee of 6 members is to be chosen from 7 men and 5 women such that the committee contains at least 2 women and at most 4 men. The number of such committees possible is:

A string of length 6 is formed by arranging the letters A, B, C, D, E, F in some order (each letter used exactly once). How many such strings are there in which A does not appear immediately before B, and C does not appear immediately before D?

Four distinct boys and four distinct girls are to be seated in a row of 8 seats such that no two boys sit next to each other and no girl occupies either of the two end seats. In how many ways can this be done?

A family of 8 people - 3 distinct brothers, 3 distinct sisters, and 2 distinct parents - stands in a row for a photograph such that the two parents stand together at one of the two ends of the row, and the 3 brothers do not all stand together. In how many such arrangements is this possible?

A bag contains 5 red balls and 4 blue balls. Two balls are drawn one after another, without replacement. Given that at least one of the two drawn balls is red, what is the probability that both balls are red?

The five distinct letters A, B, C, D, E are arranged in a row uniformly at random. What is the probability that, in the resulting arrangement, A appears before B, B appears before C, and D appears before E?

The digits 1, 2, 3, 4, 5, 6 are arranged uniformly at random to form a 6 - digit number (each digit used exactly once). What is the probability that the resulting number is divisible by 4?