Question

How many ways are there to select 4 cards from a standard deck of 52 cards (how many combinations are there)?

Answer 1

Hint: Assume the total number of cards in the standard deck as ‘n’ and consider the number of cards that is to be selected as ‘r’. Now, apply the formula of combinations for the selection of r cards from a total of n cards given as: - Number of ways to select = \[{}^{n}{{C}_{r}}\]. Simplify this relation by the expansion formula: - \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\] to get the answer.

Complete step-by-step solution:
Here, we have been provided with a standard deck of 52 cards and we have been asked to select 4 cards. We are asked to determine the total number of ways to select these 4 cards from the given 52 cards.
Now, we know that according to the combinations, if we have been provided with a total of ‘n’ number of articles and we are asked to select ‘r’ articles from them then we use the formula of combinations given as: - \[{}^{n}{{C}_{r}}\]. So, let us consider the total number of cards in the standard deck as ‘n’ and the number of cards to be selected as ‘r’. So, we have,
\[\Rightarrow \] n = 52 and r = 4
Applying the formula of combinations, we get,
\[\Rightarrow \] Number of ways to select r cards from n cards = \[{}^{n}{{C}_{r}}\]
Substituting the values of n and r, we get,
\[\Rightarrow \] Number of ways to select 4 cards from 52 cards = \[{}^{52}{{C}_{4}}\]
Using the conversion formula: - \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\], we get,
\[\Rightarrow \] Number of ways to select 4 cards from 52 cards = \[\dfrac{52!}{4!\left( 52-4 \right)!}\]
\[\Rightarrow \] Number of ways to select 4 cards from 52 cards =\[\dfrac{52!}{4!48!}\]
Converting 52! into\[52\times 51\times 50\times 49\times 48!\], we get,
\[\Rightarrow \] Number of ways to select 4 cards from 52 cards = 270,725.
Hence, there are 270,725 possible combinations for the selection of cards.

Note: One must know the difference between ‘permutations’ and ‘combinations’ to solve the above question. We use permutation when we have to arrange ‘r’ things from a total of ‘n’ things instead of selecting. Remember the formula of permutation given as: - \[{}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]. Do not get confused in the two terms, so read the question carefully if we have to select the things or arrange them. Note that you can leave the answer up to the expression \[\dfrac{52!}{4!48!}\].