Question

A supermarket offers ice cream in 10 different varieties.
Ice creams of each variety are identical in how many ways 4 ice creams can be selected:
(i) Without any restriction
(ii) At least 2 ice creams of the same variety are taken

Answer 1

Hint: In this question, for part (i), we can choose any variety for the 4 ice creams that are to be selected without any effect from the other choices and for part (ii), we have to first choose the variety of which two choices are to be made and then we can choose any variety for the remaining 2 choices without any effect from the other choices.

Complete step-by-step answer:
The most important formula that would be used in this question is as follows
The formula for calculating the total number of ways for choosing r different objects from n different objects is as follows
\[={}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]

As mentioned in the question, we have to find the total number of ways in which we can choose the ice creams while fulfilling the conditions that are mentioned in the question’s part.
Now, for part (i), the total number of ways can be calculated as follow
As mentioned in the hint, we can use the fact that in part (i), we can choose any variety for the 4 ice creams that are to be selected without any effect from the other choices. So, we can write as follows
Total number of ways
\[\begin{align}
  & ={}^{n}{{C}_{r}}\times {}^{n}{{C}_{r}}\times {}^{n}{{C}_{r}}\times {}^{n}{{C}_{r}} \\
 & ={}^{10}{{C}_{1}}\times {}^{10}{{C}_{1}}\times {}^{10}{{C}_{1}}\times {}^{10}{{C}_{1}}={{10}^{4}} \\
\end{align}\]
(Because we are given 4 choices to make and for each choice, we have 10 different options)

Now, for part (ii), the total number of ways can be calculated as follow
As mentioned in the hint, we can use the fact that in part (ii), we have to first choose the variety of which two choices are to be made and then we can choose any variety for the remaining 2 choices without any effect from the other choices. So, we can write as follows
Also, at least 2 of one type mean that there can be 2, 3 or 4 ice creams of that type. So, for choosing that variety which will be at least 2 times is as follows
\[={}^{n}{{C}_{r}}={}^{10}{{C}_{1}}=10\]
Now, we know that for the remaining two choices and for overall calculation, we can write as follows\[\]
Total number of ways
\[\begin{align}
  & =10\times {}^{n}{{C}_{r}}\times {}^{n}{{C}_{r}} \\
 & =10\times {}^{10}{{C}_{1}}\times {}^{10}{{C}_{1}}={{10}^{3}} \\
\end{align}\]
(Because we are given 4 choices to make and for each choice, we have 10 different options but at least two of them must of the same type)

Note: In the second part, it is important to notice that the other two options may or may not be of the same type as of the two choices that have to be the same at least.
It is very essential for the students to know the correct formula for calculating the number of ways in which r different objects can be chosen from n different objects which is as follows
\[={}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
In these questions, there are high chances of students wrongly assessing the question and then ending up with the wrong answer.