Question

A shop sells $6$ different flavors of ice-creams. In how many ways can a customer choose $4$ ice-cream cones if they contain only $2$ or $3$ different flavors.

Answer 1

A customer choose $4$ ice-cream cones

They contain only $2$ or $3$ different flaours.

Case 1: $6$ icecreams of $3$ different flavours are chosen

Let the flavours be $\{a,b,c,d,e,f\}$

$\Rightarrow$let flavour $a$ occur $2$ times

$\therefore$ Choose the flavours $a,a,b,c$

${=}_{C_3}^6\times \frac{3!}{2!}$

$=\frac{6!}{3!3!}\times \frac{3!}{2!}$

$=\frac{6\times 5\times 4\times 3!}{3!2!}$

$=6\times 5\times 2=60$ ways.

Case 2:two different flavours are choosen

$\Rightarrow$ two flavours repeat $2$ times.

Let $\{a,b\}$ be the repeated flavours.

$a,a,b,b$ be the flavours

${=}_{C_2}^6\times 3$

$=\frac{6!}{4!2!}\times 3$

$=\frac{6\times 5\times 4!}{4!2!}\times 3$

$=45$ times

$\therefore$ Number of ways$=2$ different flavours$+3$ different flavours

$=60+45=105$ ways.