The probability (rounded to the nearest whole percent) that a randomly selected gym member would buy passion fruit vitamin water but not tropicalgrape vitamin water is 17 percent.
5x - 3y + 1 = 16 ⇒ 10x - 6y + 2 = 16
2x + 5y + 0 = -6 ⇒ <u>10x + 25y + 0 = -6</u>
31y + 2 = 10
<u> -2 -2</u>
<u>31y</u> = <u>8</u>
31 31
y = 8/31
5x - 3(8/31) + 1 = 16
5x - 24/31 + 1 = 16
5x + 7/31 = 16
<u> -7/31 -7/31</u>
<u>5x</u> = <u>10 24/31</u>
5 5
x = 2 24/155
(x, y) = (2 24/155, 8/31)
When you have 3 choices for each of 6 spins, the number of possible "words" is
3^6 = 729
The number of permutations of 6 things that are 3 groups of 2 is
6!/(2!×2!×2!) = 720/8 = 90
A) The probability of a word containing two of each of the letters is 90/729 = 10/81
The number of permutations of 6 things from two groups of different sizes is
(2 and 4) : 6!/(2!×4!) = 15
(3 and 3) : 6!/(3!×3!) = 20
(4 and 2) : 15
(5 and 1) : 6
(6 and 0) : 1
B) The number of ways there can be at least 2 "a"s and no "b"s is
15 + 20 + 15 + 6 + 1 = 57
The probability of a word containing at least 2 "a"s and no "b"s is 57/729 = 19/243.
_____
These numbers were verified by listing all possibilities and actually counting the ones that met your requirements.