Question

In: Statistics and Probability

How many ways are there to order the 26 letters of the alphabet so that no two of the vowels a, e, i, o,u appear consecutively and the last letter in the ordering is not a vowel?

(a) How many ways are there to order the 26 letters of the alphabet so that no two of the vowels a, e, i, o,u appear consecutively and the last letter in the ordering is not a vowel?

Hint: Every vowel appears to the left of a consonant.

(b) How many ways are there to order the 26 letters of the alphabet so that there are at least two consonants

immediately following each vowel?

(c) In how many different ways can 2n students be paired up?

(d) Two n-digit sequences of digits 0,1,...,9 are said to be of the same type if the digits of one are a permutation of the digits of the other. For n D 8, for example, the sequences 03088929 and 00238899are the same type. How many types of n-digit sequences are there?

Expert Solution

(a)

We have 21 consonants and 5 vowels. We have been told that we cannot have consecutive vowels and the last letter should not be a vowel. Also all vowels are on the left of the consonant.

bcdfghjklmnpqrstvwxyz

b_c_d_f_g_h_j_k_l_m_n_p_q_r_s_t_v_w_x_y_z

In all we have 26 alphabets. Therefore,

21! ways to order the consonants.

Now we count the blanks between each consonant since remember 1st and last cannot be a vowel we will have 19 blanks for vowels. Therefore

19P5 ways to order the vowels in between the consonants.

Final Ans = 21! * 19P5

(b)

We still have 21! ways to order consonants but the number of blanks in between will reduce.

b_cd_fg_hj_kl_mn_pq_rs_tv_wx_yz

We have 10 blanks to order the vowels now

10P5 ways to order vowels now

Final Ans = 21! * 10P5

(c)

If there are 2n students to paired that means we have 2 blanks for 2n students to be filled

Final Ans = 2nP2 or ( 2n * 2n-1 )

(d)

Apologies but I am unable to solve for this question. Although using 'stars and bars' concept we can arrive at a solution. I'll try to solve it further.

orchestra answered 2 years ago

The English alphabet has 26 letters. There are 6 vowels. (a, e, i, o, u, and...

The English alphabet has 26 letters. There are 6 vowels. (a, e, i, o, u, and sometimes y). Suppose we randomly select 8 letters from the alphabet without replacement. Let X = the number of vowels chosen (including y as a vowel). a. How many possible ways are there to select the 8 out of 26 letters (order does not matter) without replacement? b. What is the probability that X = 2 c. What is the probability that X=1? d....

4. The Hawaiian alphabet has twelve letters: five vowels (a, e, i, o, and u) and...

4. The Hawaiian alphabet has twelve letters: five vowels (a, e, i, o, and u) and seven consonants (h, k, l, m, n, p, and w). For the purpose of this exercise we will define an n–letter “word” as an ordered collection of n of these twelve letters with repeats allowed. Obviously, most such “words” will be nonsense words. a) How many possible four–letter Hawaiian “words” are there? b) What is the probability a randomly selected four–letter “word” has no...

Question 13 We define the letters 'a', 'e', 'i', 'o' and 'u' as vowels. We do...

Question 13 We define the letters 'a', 'e', 'i', 'o' and 'u' as vowels. We do not consider any other letter as a vowel. Write a function named initialVowels() that returns a list of words in a body of text that begin with a vowel. Include both capitalized and lower case instances of the vowels. A word should appear in the return list at most once, no matter how many times is occurs in the input string. Input: a string...

How many ways are there to arrange the letters ‘a’, ‘b’, ‘c’, ‘d’, and ‘e’ such...

How many ways are there to arrange the letters ‘a’, ‘b’, ‘c’, ‘d’, and ‘e’ such that ‘a’ is not immediately followed by ‘e’ (no repeats since it is an arrangement)? Justify your answer using the product rule, the sum rule, and/or the subtraction rule .

Assuming that each letter can appear at most once in an arrangement, how many ways are...

Assuming that each letter can appear at most once in an arrangement, how many ways are there to arrange 10 letters taken from the alphabet a-z such that: a) Both a and z are included in the arrangement? b) Exactly one of a and z appears in the arrangement? The intention here is that exactly one element from the set {a,z} appears in the string. So, abcdefghij and bcdezfghij are OK, but abcdezfghi is not. c) z and a appear...

In how many ways can you rearrange the letters A, B, C, D, E?

Let S denote the 10-element set {a,b,c,d,e,f,g,h,i,j}. How many ways can we construct a subset of...

Let S denote the 10-element set {a,b,c,d,e,f,g,h,i,j}. How many ways can we construct a subset of S of size 7 ? 120 How many ways can we construct a subset of S of size 7 containing the element j? 84 How many ways can we construct a subset of S of size 7 containing i but not j ? 28 How many ways can we construct a subset of S of size 7 containing h but neither i nor j...