Question

In how many different orders can six colored blocks be chosen from a set of 23 different blocks?

Answer 1

Use the formula of combinations given as ${}^{n}{{C}_{r}}$ to find the numbers of possible ways to select r things out of a total of n things. Consider n as the total number of different colored blocks present and r as the number of blocks required to be selected. Now, use the formula $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ to simplify the expression and get the answer in number.

Complete step-by-step solution:
Here we have been given that there is a set of 23 different color blocks in which 6 colored blocks needs to be selected. We are asked to determine the total number of ways in which the selection can be done.
Now, we know that if we have to select r number of things from a total of n number of things then we use the formula of combination given as ${}^{n}{{C}_{r}}$. So let us consider the total number of blocks as n and the number of blocks to be selected as r, so we have n = 23 and r = 6, substituting these values in the formula we get,
$\Rightarrow$ Number of possible ways of selection = ${}^{n}{{C}_{r}}={}^{23}{{C}_{6}}$
Using the formula $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ we get,
$\Rightarrow$ Number of possible ways of selection = $\dfrac{23!}{6!\left( 23-6 \right)!}$
$\Rightarrow$ Number of possible ways of selection = $\dfrac{23!}{6!17!}$
We can write $23!=23\times 22\times 21\times 20\times 19\times 18\times 17!$, so we get,
$\Rightarrow$ Number of possible ways of selection = $\dfrac{23\times 22\times 21\times 20\times 19\times 18}{6\times 5\times 4\times 3\times 2\times 1}$
$\Rightarrow$ Number of possible ways of selection = 5313
Hence, total number of ways to select the required number of blocks is 5313.

Note: Do not use the formula of permutation because we are not asked to arrange these students but we have to select them only. You must know the difference between the two terms ‘permutation’ and ‘combination’. In general if we have to select r things from n thing then we apply the combinations formula and if after selection we need to arrange those things also then we apply permutation formula given as ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.