Find the number of homogeneous products of r dimensions that can be formed out of the n letters a, b, c ….and their powers.
By division, or by the binomial theorem, we have:
Hence, by multiplication,
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.
becomes , or
Hence, the coefficient of in the expansion of
Find the number of terms in the expansion of any multinomial when the index is a positive integer.
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:
Find the coefficient of in the expansion of
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
Find the value of the series
The expression is equal to
If n is any positive integer, show that the integral part of is an odd number.
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,