With sachets containing at most 13 seeds, given 2021 rose, jasmine or fennel seeds respectively what is the maximum number of rose seeds used?

Question

I have taken a part of the problem which I did not understand its solution.

Basically, there are: $2021$ rose seeds, $2021$ jasmine seeds and $2021$ fennel seeds.

So now, with sachets of at most size 13 (meaning each sachet can contain 1 seed or 2 seeds or 3 seeds or ... or 12 seeds or 13 seeds), what is the [maximum] number of rose buds which can be used?

I believe the official solution is counting them like this:

$$C_3^3 + C_4^3 + C_5^3 + \ldots+ C_{14}^3 + C_{15}^3 = C_4^{16}$$

The problem I am having is: how does the answer count on the Left Hand Side (LHS)?

Please note that I understand $C_r^n$, which is the Binomial Coefficient as well as that summation as $C_4^{16}$, which is the "Hockey Stick Identity".

The Original Problem is like the following:

Carmilla is making sachets by placing dry rose, jasmine buds and fennel seeds into cloth bag. If each sachet contains at least one rose, jasmine bud or fennel seed, and Carmilla has $2021$ rose, $2021$ jasmine buds and $2021$ fennel seeds in hand, and NO two sachets contain the same number of rose AND jasmine AND fennel seeds at the same time. What is the maximum number of sachets she can make?

Could someone explain the part of the solution where I have highlighted?

Answer 1

$$C_3^3 + C_3^4 + C_3^5 + \ldots + C_3^{14} + C_3^{15} = C_4^{16}$$

The meaning of term $C_3^{s+2}$, $1\le s\le 13$ is the number of rose buds in all (different) sachets of size $s$ since $C_3^{s+2}=\frac s3\times C_2^{s+2}$. More specifically,

1. There are $C_2^{s+2}$ (different) sachets of size $s$.
2. There are $sC_2^{s+2}$ items in all $C_2^{s+2}$ sachets of size $s$.
3. Because of symmetry, exactly $\frac13$ of all those items are rose buds. So the number of rose buds among those items is $\frac13sC_2^s=\frac s3\frac{(s+2)(s+1)}{1\cdot2}=C_3^{s+2}$.

---

There is slight ambiguity whether Carmilla has 2021 roses or 2021 rose buds. This answer uses "rose buds".