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I have a very basic and conceptual doubt regarding the function of the permutation formula. I am going to ask them through some examples in which I applied them and could not proceed.

The first basic thing is that let us say I have four dice and I want two of my dice to show exactly $3$ and the rest can show any number except $3$.

Now my line of thought was that we fix two places where we fill in $3$ and the rest two places can be filled in $5$ ways each. the first die can show in $5$ ways, the second can also show $5$ ways, the third and fourth can show in $1$ ways each, so the total number of arrangement should be $5 \cdot 5 \cdot1 \cdot 1$. My first doubt is - Does this thing need arrangement and rearrangement or is this automatically arranged? If not then how do I write it using the permutation formula so that the direct result which comes out is automatically arranged and I don't need to look for ways to arrange it.

Harsh Sharma
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1 Answers1

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You need arrangement. Because two 3's can be on any two places. You can multiply your ways with C(4,2). Here r = 2 because you have two 3's.

So answer will be $C(4,2) * (5*5*1*1)$

In case you want to use permutation. Then,

Number of ways = $\frac{P(4,2) * (5 * 5 * 1 * 1)}{2}$

Dividing by 2 because 3 is repeating twice.

Kanwaljit Singh
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