Exercise 2.4

 (10) How many ways can the letters of the word 'BANANA' be arranged so that 

(i) all A's come together

(ii) no two A's come together

In the word 
BANANA
total no.of letters =6
no.of A's =3 
NO.OF n's = 2

(i)all A's come together

Consider (AAA) as one unit

(A,A,A),N,N,B can be arranged in $\frac{4!}{2!}$=12 ways

(ii)

no two A 's come togeter

---B--N--N--- can be arranged in $\frac{3!}{2!}$=3
there are 4 places for arrangeing 3 A 's
A,A,A cna be arranged in 4 places in $\frac{{}^{4}P_3}{3!}=4$ ways
no.of permutations =3 x 4 =12