Question

Find the number of arrangements of the letters of the word MADHUBANI. in how many of these arrangements
(i) Do the words starts with $B$?
(ii) Do all the vowels never occur together?
(ii) Do the words begin with $D$ and end in $H$?

Answer 1

MADHUBANI

$A-2B-1D-1H-1I-1M-1N-1U-1$

Total arrangement $9!\times \frac{1}{2!}=\frac{9!}{2!}$

i) No. of words starts with $8=\frac{8!}{2!}=20!60$

ii) All the vowels never occur together

${}_{(}A{)}_{(}A{)}_{(}I{)}_{(}U)$

Total words $=\frac{4!}{2!}\times {}^5{C}_3\times {}^5{C}_2\times 5!$

$=12\times 120\times 10\times 10$

$=144000$

No of words begin with $D$

are end with $H=\frac{7!}{2!}=2520$