In a pack of 42 cards, 3 cards are chosen one after the other. Find the number of ways this can be done without replacement:

Correct option is C

When selecting items one after another without replacement and where the order of selection matters, permutations are used. In this case, there are 42 cards in total (n = 42), and 3 cards are chosen (r = 3).
The formula for permutations without replacement is:
​$$\mathrm{P(n, r) = \frac{n!}{(n-r)!}}$$​​
where:
$$\text{n}$$​ is the total number of item.
$$\text{r}$$​ is the number of items to choose and arrange
$$\text{!}$$​ denotes the factorial function (e.g., $$5!=5\times 4\times 3\times 2\times 1$$​)
Substituting the values into the formula:
​$$P(42, 3)=\frac{42!}{(42-3)!}$$​​
​$$P(42, 3)=\frac{42!}{39!}$$​​
The factorials can be expanded:
​$$P(42, 3)=\frac{42\times 41\times 40\times 39\times 38\times ...\times 1}{39\times 38\times ...\times 1}$$​​
The 39! terms in the numerator and denominator cancel out, leaving:
​$$P(42, 3)=42\times 41\times 40$$​​
​$$P(42, 3)=68880$$​​
Therefore, there are 68,880 ways to choose 3 cards one after the other from a pack of 42 cards without replacement.
Information Booster:
1. Permutation without replacement:
· Used when the order of selection matters, and no item is selected more than once.
· Formula: P(n, r) = n × (n−1) × (n−2)... up to r terms.