Question

There are $n$ different gift coupons, each of which can occupy $N ( N > n )$ different envelopes, with the same probability $1 / N$
$P _1$ : The probability that there will be one gift coupon in each of n definite envelopes out of $N$ given envelopes
$P _2$ : The probability that there will be one gift coupon in each of $n$ arbitrary envelopes out of $N$ given envelopes Consider the following statements
(i) $P _1= P _2$
(ii) $P _1=\frac{ n !}{ N ^{ n }}$
(iii) $P _2=\frac{ N !}{ N ^{ n }( N - n ) !}$
(iv) $P _2=\frac{ n !}{ N ^{ n }( N - n ) \text { ! }}$
(v) $P _1=\frac{ N \text { ! }}{ N ^{ n }}$
Now, which of the following is true

• A. Only (i)
• B. (ii) and (iii)
• C. (ii) and (iv)
• D. (iii) and (v)

Answer 1

Answer: (B)

From the given data $n(S)=N^n ; n(A)=n ! \Rightarrow P_1=\frac{n !}{N^n}$
$P _1=\frac{ n !}{ N ^{ n }}$ Since the $n$ different gift coupans can be placed in the $n$ definite (Out of $N$ ) envelope in ${ }^{ n } P _{ n }= n$ ! ways
$P _2=\frac{ N !}{( N - n ) ! N ^{ n }}$ As $n$ arbitrary envelopes out of $N$ given envelopes can be chosen in ${ }^{ N } C _{ n }$ ways and the $n$ gift coupans can occupy these envelopes in $n$ ! ways. ]
$=\frac{ N !}{ N ^{ n }( N - n ) !}$