OGF

In how many ways a total of 16 can be obtained by rolling 4 dice? Solve using OGF.

#Combinatorics

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Agnishom Chattopadhyay

Ashley Shamidha - 3 years, 7 months ago

That would be the coefficient of $x^{16}$ in $(x + x^2 + x^3 + x^4 + x^5 + x^6)^4$ , right?

Agnishom Chattopadhyay - 3 years, 7 months ago

The answer is right but how did u arrive at it

Ashley Shamidha - 3 years, 7 months ago

@Ashley Shamidha – Well, I probably will not be able to give a very good explanation. But think about this:

What is the coefficient of $x^n$ in $(a_0 + a_1 x + a_2 x^2 + \cdots a_k x^k) (b_0 + b_1 x + b_2 x^2 + \cdots b_k x^k)$ ?

It is precisely the sum of products of pairs $a_i b_j$ such that $i + j = n$ .

If this makes sense, then I can proceed further. Let me know.

Agnishom Chattopadhyay - 3 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$