Question

Out of all the arrangements that can be made taking 5 letters at a time of the word BRILLIANT one is chosen at random. The probability that this will have 5 distinct letters is

• A. $\frac{255}{702}$
• B. $\frac{257}{502}$
• C. $\frac{252}{507}$
• D. $\frac{522}{705}$

Answer 1

The correct option is C $\frac{252}{507}$

The given word is made up of the letters $LL,II,B,R,A,N,T$
Possibilities of arrangements: Number of arrangements:
(i) two alike + two alike + one different $1\times 5\times \frac{5!}{(2!{)}^2}=150$
(ii) two alike + three different $2\times {}^6{C}_3\times \frac{5!}{2!}=2400$
(iii) all are different ${}^7{C}_5\times 5!=2520$
Total number of different arrangements $=150+2400+2520=5070$
The required probability $=\frac{2520}{5070}=\frac{252}{507}$