The number of words of four letters containing equal number of vowels and consonants, repetition being allowed, is
A.
$${105^2}$$
B.
$$210 \times 243$$
C.
$$105 \times 243$$
D.
None of these
Answer :
$$210 \times 243$$
Solution :
The number of selections of 1 pair of vowels and 1 pair of consonants $$ = {\,^5}{C_1} \times {\,^{21}}{C_1}.$$
The number of selections of 2 different vowels and 2 different consonants $$ = {\,^5}{C_2} \times {\,^{21}}{C_2}.$$
∴ the required number of words
$$ = {\,^5}{C_1} \times {\,^{21}}{C_1} \times \frac{{4!}}{{2!\,2!}} + {\,^5}{C_2} \times {\,^{21}}{C_2} \times 4!.$$
Releted MCQ Question on Algebra >> Permutation and Combination
Releted Question 1
$$^n{C_{r - 1}} = 36,{\,^n}{C_r} = 84$$ and $$^n{C_{r + 1}} = 126,$$ then $$r$$ is:
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 ; and then the men select the chairs from amongst the remaining. The number of possible arrangements is