When the following expression is expanded, what will be the coefficient of ?

Source: mathcontest.olemiss.edu 7/7/2008

**SOLUTION**

**STEP 1 ****Product of two factors**:

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Let be the 9 coefficients, then we have

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Let . The coefficients follow the pattern

*A 2A A 4A 8A 4A 4A 8A 4A*.

**STEP 2 ****Product of three factors**:

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The product is expanded as follows:

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The 27 coefficients follow the pattern

**STEP 3 ****Product of four factors**:

The product generates 81 coefficients and is too long to expand here. So, we will represent them in 3 groups of 27 each as follows:

**STEP 4 ****Try solving a simpler problem**: What is the value of ?

Since belongs to the group , we write

where is the coefficient in that corresponds to .

Given that corresponds to , .

Thus, .

STEP 5 **Try solving another simpler problem**: What is the value of ?

Since the previous product produces only 81 coefficients , we need to go further to find the value of . We need to find the product of five factors which will produce coefficients .The new factor is

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Again, we divide the 243 coefficients into 3 groups of 81 each as follows.

Since belongs to the group , we write

where is the coefficient in that corresponds to .

Given that corresponds to .

Thus, .

We need to go back to **STEP 3** to find the value of .

Remember in **STEP 3** we found 81 coefficients arranged as follows:

Since belongs to the group , we write

where is the coefficient in that corresponds to .

Given that corresponds to .

Thus, .

But, how much is ?

From **STEP 2**, .

So, .

Finally, .

**STEP 6 ****What is the value of** ?

Now we are ready to tackle the big problem of finding the value of .

The product of 7 factors generates coefficients . The seventh factor is

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We arrange in 3 groups of 729 each as follows:

Since belongs to the third group and corresponds to , we write

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The product of 6 factors generates coefficients . The sixth factor is

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We arrange in 3 groups of 243 each as follows:

The above arrangement tells us that .

From **STEP 5**, we have

Thus, .

Now, we can put everything together one calculation at a time.

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**Answer**: .