OlympiadEasyCombinatorics10–12

After like terms are collected, how many different terms does the expansion of $(x + y + z)^{4}$ have?

Depois de reduzidos os termos semelhantes, quantos termos diferentes tem o desenvolvimento de $(x + y + z)^{4}$ ?

Answer options

Solution

A term is $x^{a}y^{b}z^{c}$ with $a + b + c = 4$ and $a, b, c \ge 0$ . Stars and bars counts these as $\binom{4 + 2}{2} = \binom{6}{2} = 15$ . The $81 = 3^{4}$ counts the products before collecting.

After like terms are collected, how many different terms does the expansion of (x+y+z)4(x + y + z)^{4}(x+y+z)4 have?

Answer options

After like terms are collected, how many different terms does the expansion of $(x + y + z)^{4}$ have?