Question

Problem 6: A baker has n cupcakes in a line. They break the line into some number of sublines, then they put two (identical) cherries on two cupcakes from each subline, and finally a quality control robot marks each subline as good or bad. Let fn be the number of ways this can happen. Find the ordinary generating function of f

combinatorics

Answer 1

ANSWER :-

A baker has cupcakes in a line = n

They put two (identical) cherries on two cupcakes from each sub-line.

Let $f_n$ be the number of ways this can happen.

Here we need to find the ordinary function of $f_n$ :-

By the use of two cupcakes from each sub-line.

the function is

n.C2

Therefore the ordinary generating function

rl 7t

rl 7t

we already know that the formula for

n.C2

$\Rightarrow \frac{(n)!}{(n-2)!2!}$

(n - 2) (n 1)2!

By cancellation of (n-1)(n-2) in numerator and denominator

$\Rightarrow \frac{(n)}{2!}$

2

$\Rightarrow \frac{n}{2}$

Since, in n/2 no.of ways this can happen.

Finally,we found ordinary generating function and no.of ways that happens.

Thank you