The number of ways of distributing $15$ identical balloons, $6$ identical pencils and $3$ identical erasers among $3$ children, such that each child gets at least four balloons and one pencil, is

We need to distribute 15 identical balloons, 6 identical pencils, and 3 identical erasers among 3 children, with each child getting at least 4 balloons and 1 pencil.

---

Since each child must get at least 4 balloons and 1 pencil, let's distribute these minimum amounts first:

Give to each child:

4 balloons

1 pencil

Items used:

Balloons: 3 children × 4 balloons = 12 balloons

Pencils: 3 children × 1 pencil = 3 pencils

Remaining items to distribute freely:

Balloons: 15 - 12 = 3 balloons

Pencils: 6 - 3 = 3 pencils

Erasers: 3 erasers (no minimum constraint)

---

Now we need to distribute the remaining items. Since items of the same type are identical, we use the Stars and Bars method.

When distributing n identical items among k people, we're finding how many ways we can write:

$x_1 + x_2 + x_3 = n$ (where each $x_i \geq 0$)

If we have 3 items and want to divide them among 3 people, we need 2 dividers to create 3 groups:

one arrangement gives person 1 two items, person 2 one item, person 3 zero items

another arrangement gives person 1 one item, person 2 zero items, person 3 two items

The formula for n identical items distributed among k people:

Number of ways = ${{}}^{{n+k-1}}C_{{k-1}} = {{}}^{{n+k-1}}C_{{n}}$

We arrange n items and (k-1) separators in a line. Total positions = n + (k-1). We choose (k-1) positions for separators from these total positions.

---

For 3 remaining balloons among 3 children:

n = 3 (balloons), k = 3 (children)

Ways = ${{}}^{{3+3-1}}C_{{3-1}} = {{}}^{{5}}C_{{2}} = \dfrac{5!}{2! \times 3!} = 10$ ways

For 3 remaining pencils among 3 children:

n = 3 (pencils), k = 3 (children)

Ways = ${{}}^{{5}}C_{{2}} = 10$ ways

For 3 erasers among 3 children:

n = 3 (erasers), k = 3 (children)

Ways = ${{}}^{{5}}C_{{2}} = 10$ ways

---

Since the distribution of balloons, pencils, and erasers are independent events, we multiply the number of ways:

Total ways = $10 \times 10 \times 10 = 1000$

---

When distributing multiple types of identical items independently, we multiply the number of ways for each type. This is because for each way of distributing balloons, we have all possible ways of distributing pencils, and for each combination of balloon-pencil distribution, we have all possible ways of distributing erasers.

Answer: 1000