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(B) To find the number of ways such that exactly $4$ cards go into their respective envelopes:$1$. First, select $4$ cards out of $7$ to be placed in

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$2 	imes { }^{7} C_{3}$

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(B) To find the number of ways such that exactly $4$ cards go into their respective envelopes:$1$. First, select $4$ cards out of $7$ to be placed in their correct envelopes. This can be done in ${ }^{7} C_{4}$ ways.$2$. The remaining $3$ cards must be placed in the remaining $3$ envelopes such that none of them go into their respective envelopes (this is a derangement of $3$ objects, denoted by $D(3)$).$3$. The number of derangements of $n$ objects is given by $D(n) = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!}ight)$.$4$. For $n = 3$, $D(3) = 3! \left(1 - 1 + \frac{1}{2} - \frac{1}{6}ight) = 6 	imes \left(\frac{1}{3}ight) = 2$.$5$. Total ways = ${ }^{7} C_{4} 	imes D(3) = { }^{7} C_{3} 	imes 2 = 2 	imes { }^{7} C_{3}$.

There are $7$ greeting cards, each of a different colour, and $7$ envelopes of the same $7$ colours as the cards. The number of ways in which the cards can be put in envelopes, so that exactly $4$ of the cards go into envelopes of the respective colour, is:

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