Homology of product of spheres

Statement

Let $(m_1,m_2,\dots ,m_r)$ be a tuple of nonnegative integers. Let $A$ be the space $S^{m_1}\times S^{m_2}\times S^{m_3}\times \dots \times S^{m_r}$. Then the homologies of $A$ are free Abelian, and the $q^{th}$ Betti number is given by the following formula:

$b_q(A)=\left|\{T\subset \{1,2,3,\dots ,r\}|{\sum }_{i\in T}{m}_i=q\}\right|$

In other words $b_q(A)$ is the number of ways $q$ can be obtained by summing up subsets of $(m_1,m_2,\dots ,m_r)$.

A particular case of this is when all the $m_i$s are 1. In this case:

$(\genfrac{}{}{0ex}{}{b_q(A)=r}{q})$.

Proof

Using exact sequence for join and product

Further information: exact sequence for join and product

The above claim can be easily proved using induction, and the exact sequence for join and product.