**Question:**

Find the number of homogeneous products of r dimensions that can be formed out of the n letters a, b, c ….and their powers.

**Solution:**

By division, or by the binomial theorem, we have:

Hence, by multiplication,

suppose;

where , , , are the sums of the homogeneous products of one, two, three, … dimensions that can be formed of a, b, c, …and their powers.

To obtain the number of these products, put a, b, c, …each equal to 1; each term in , , , …now becomes 1, and the values of , , , …so obtained give the number of the homogeneous products of one, two, three, ….dimensions.

Also,

becomes , or

Hence, the coefficient of in the expansion of

**Question:**

Find the number of terms in the expansion of any multinomial when the index is a positive integer.

**Answer:**

In the expansion of

every term is of n dimensions; therefore, the number of terms is the same as the number of homogeneous products of n dimensions that can be formed out of the r quantities , , , …, and their powers; and therefore by the preceding question and solution, this is equal to

**A theorem in combinatorics:**

From the previous discussion in this blog article, we can deduce a theorem relating to the number of combinations of n things.

Consider n letters a, b, c, d, ….; then, if we were to write down all the homogeneous products of r dimensions, which can be formed of these letters and their powers, every such product would represent one of the combinations, r at a time, of the n letters, when any one of the letters might occur once, twice, thrice, …up to r times.

Therefore, the number of combinations of n things r at a time when repetitions are allowed is equal to the number of homogeneous products of r dimensions which can be formed out of n letters, and therefore equal to , or .

That is, the number of combinations of n things r at a time when repetitions are allowed is equal to the number of combinations of things r at a time when repetitions are NOT allowed.

*We conclude this article with a few miscellaneous examples:*

**Example 1:**

Find the coefficient of in the expansion of

**Solution 1:**

The expression , suppose.

The coefficients of will be obtained by multiplying , , by 1, -4, and 4 respectively, and adding the results; hence,

the required coefficient is

But, with a little work, we can show that .

Hence, the required coefficient is

**Example 2:**

Find the value of the series

**Solution 2:**

The expression is equal to

.

**Example 3:**

If n is any positive integer, show that the integral part of is an odd number.

**Solution 3:**

Suppose I to denote the integral and f the fractional part of .

Then, …call this relation 1.

Now, is positive and less than 1, therefore is a proper fraction; denote it by ;

Hence, …call this as relation 2.

Add together relations 1 and 2; the irrational terms disappear, and we have

But, since f and are proper fractions their sum must be 1;

Hence, I is an odd integer.

*Hope you had fun,*

*Nalin Pithwa.*