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Q.
If $4$ letter words are formed using letters of the word ‘MORADABAD’, then the probability that D comes exactly once in the $4$ letter word is
NTA AbhyasNTA Abhyas 2020Probability
Solution:
Total cases $=4$ letter word is formed by using either $3$ identical letter and $1$ distinct letter or $2$ alike of one kind and $2$ alike of $2^{n d}$ kind or $2$ alike, $2$ distinct or $4$ distinct
Number of ways for $3$ alike and $1$ distinct $=\_{}^{1}C_{1}\times \_{}^{5}C_{1}\times \frac{4 !}{3 !}=20$
Number of ways for $2$ alike and $2$ alike $=\_{}^{2}C_{2}\times \frac{4 !}{2 ! 2 !}=6$
Number of ways for $2$ alike and $2$ distinct $=\_{}^{2}C_{1}\times \_{}^{5}C_{2}\times \frac{4 !}{2 !}=240$
Number of ways for $4$ distinct $=\_{}^{6}C_{4}\times 4!=360$
The total number of ways $=20+6+240+360=626$
For favorable cases, the following possibilities are there (one D is always included)
Number of ways for $3$ alike $=\_{}^{1}C_{1}^{}\times \frac{4 !}{3 !}=4$
Number of ways for $2$ alike $+1$ distinct $=\_{}^{1}C_{1}^{}\times \_{}^{4}C_{1}^{}\times \frac{4 !}{2 !}=48$
Number of ways for $3$ distinct $=\_{}^{5}C_{3}^{}\times 4!=240$
The total number of ways for favorable cases $=240+48+4=292$
Hence, the required probability $=\frac{292}{626}=\frac{146}{313}$