select the correct answer from each drop - down menu. a code is formed using four of these letters: a, b, i…

select the correct answer from each drop - down menu. a code is formed using four of these letters: a, b, i, k, n, o, and t. (note that the order of the letters in the code matters; for example, abik is a different code from baik.) the size of the sample space is. the probability that the first three letters of the four - letter code are vowels is.

select the correct answer from each drop - down menu. a code is formed using four of these letters: a, b, i, k, n, o, and t. (note that the order of the letters in the code matters; for example, abik is a different code from baik.) the size of the sample space is. the probability that the first three letters of the four - letter code are vowels is.

Answer

Explanation:

Step1: Calculate sample - space size

We are choosing 4 letters out of 7 where order matters. Use the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, with $n = 7$ and $r=4$. $P(7,4)=\frac{7!}{(7 - 4)!}=\frac{7!}{3!}=\frac{7\times6\times5\times4\times3!}{3!}=840$.

Step2: Calculate favorable cases

The vowels among the given letters are A and I. For the first three - letter to be vowels, the first three positions have 2 choices each (either A or I) and the fourth position has 7 choices. So the number of favorable cases is $2\times2\times2\times7 = 56$.

Step3: Calculate probability

Probability $P=\frac{\text{Number of favorable cases}}{\text{Size of sample space}}=\frac{56}{840}=\frac{1}{15}$.

Answer:

The size of the sample space is 840. The probability that the first three letters of the four - letter code are vowels is $\frac{1}{15}$.