Simplifying Product Terms

The problem asks us to find the number of terms containing exactly one '$x$' variable and the rest '$y$' variables after expanding the product: $(x_1+y_1)(x_2 + y_2) \dots (x_{10}+y_{10})$

Understanding Term Formation

When expanding this product, each resulting term is formed by selecting either '$x_i$' or '$y_i$' from each factor $(x_i+y_i)$ and multiplying the selections together. The product consists of 10 such factors.

Identifying Single 'x' Terms

We are looking for terms with exactly one '$x$' and nine '$y$'s. This means from one specific factor $(x_i+y_i)$, we must choose '$x_i$', and from all the other 9 factors, we must choose '$y_j$'.

Let's consider the possibilities:

• Choose '$x_1$' from $(x_1+y_1)$ and '$y_j$' from all other factors $(x_j+y_j)$ for $j = 2, 3, \dots, 10$. This gives the term: $x_1 y_2 y_3 \dots y_{10}$.
• Choose '$x_2$' from $(x_2+y_2)$ and '$y_j$' from all other factors $(x_j+y_j)$ for $j = 1, 3, \dots, 10$. This gives the term: $y_1 x_2 y_3 \dots y_{10}$.
• ...and so on...
• Choose '$x_{10}$' from $(x_{10}+y_{10})$ and '$y_j$' from all other factors $(x_j+y_j)$ for $j = 1, 2, \dots, 9$. This gives the term: $y_1 y_2 \dots y_9 x_{10}$.

Counting the Specific Terms

Each unique term with a single '$x$' corresponds to choosing which of the 10 factors will contribute the '$x$' variable. Since there are 10 factors, there are 10 distinct choices for the single '$x$'.

Therefore, there are 10 possible terms that have exactly one '$x$' and the rest '$y$'s.

Final Answer Determination

The number of terms with only a single '$x$' and the rest '$y$'s is equal to the number of factors, which is 10.