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?

(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?

Solutions

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 1 year ago

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