Question

In a bag there is a minimum of six old Indian coins of every denomination (i.e., Athanni, Chavanni, Dunanni, Ekanni). Number of ways in which one can take 6 coins from the bag is.

Answer 1

Hint: In this question first, we will see how many coins we need to take out from the bag. So, here according to this question we have to pick 6 coins from the bag. Then we will see how many types of denomination we have in the bag. As we can see we have four types of denominations i.e., Athanni, Chavanni, Dunanni, Ekanni. Then we will try to find the number of ways in which one can take 6 coins from the bag.

Complete step by step solution:
Number of coins one can take out from the bag $= 6$
Total number of denomination of coins $= 4$
Minimum number of coins of every denomination $= 6$
Total minimum number of four types of coins of every denomination $= 6 + 6 + 6 + 6 = 24$
Combination Formula: When r things are to be selected out of n types of things where each object may be repeated any number of times, then number of possible ways to select is given by $^{n + r - 1}{C_r}$
Here, $r = 6$ and $n = 4$
Number of ways in which one can take 6 coins from the bag = $^{n + r - 1}{C_r}$
$
  { = ^{4 + 6 - 1}}{C_6} \\
   = \dfrac{{9 \times 8 \times 7 \times 6!}}{{3! \times 6!}} = \dfrac{{9 \times 8 \times 7 \times 6!}}{{3 \times 2 \times 1 \times 6!}} = 84 \\
$

$\therefore$ The number of ways in which one can take 6 coins from the bag is 84.

Note:
In these types of questions we will use the concept of combination. Here, we will first consider that $n$ is equal to how many coins one can take out from the bag, that is, 6 according to the given question. Then we will consider $r$ which is equal to the total number of denominations we have in the bag that is 4. Then we will put these values in the above formula.