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Permutations

As head of Brilliant's cybersecurity, you need to know how many different passwords can be created by rearranging the digits 1, 2, 3, 4, 5, and 6. Can you count them all? See more

Level 4

         

How many 5 digit numbers can be formed if either
(a) every successive digit exceeds its predecessor, or
(b) every successive digit is smaller than its predecessor.

Detail: Leading 0's don't count, aka 04521 is not a 5-digit number.

In how many ways can 4 boys (A, B, C, D) and 4 girls (W, X, Y, Z) be seated in a row, such that all the girls do not sit together?

Suppose \(3\) married couples are randomly seated around a circular table. (It is not required that the members of any couple must be seated together.) The probability that no spouses are seated next to one another is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

Alice, Benedict, Calvin, Dennis and Evelyn attended a party. At the end of the night, they each randomly grabbed a left shoe and a right shoe. What is the probability that each person left with exactly one of their own shoes?

Note: If the probability is 1%, enter it as 0.01.

\[\huge INTERMEDIATE\] How many words can be made with the letters of the word \(INTERMEDIATE\) if the order of vowels doesn't change ?

The "order of the vowels" refers to the sequence in which they occur. IE the sequence should be \( I - E - E - I - A - E \).

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