Question 2

We need to find all the ways to place '+' or '-' signs between four consecutive numbers.

Step 1 — Understanding the problem

We are given four consecutive numbers, like 3, 4, 5, and 6. We need to put either a '+' sign or a '-' sign in the spaces between these numbers. There are three spaces where we can place these signs.

3 \text{ _ } 4 \text{ _ } 5 \text{ _ } 6

Step 2 — Calculating the number of possibilities

For each of the three spaces, we have two choices. We can choose either a '+' sign or a '-' sign for the first space. We can choose either a '+' sign or a '-' sign for the second space. We can choose either a '+' sign or a '-' sign for the third space. To find the total number of different ways, we multiply the number of choices for each space.

$\text{Total possibilities} = \text{Choices for 1st space} \times \text{Choices for 2nd space} \times \text{Choices for 3rd space}$

$= 2 \times 2 \times 2$

$= 2^3$

$\boxed{8 \text{ possibilities}}$

Step 3 — Listing all possibilities

Let us systematically list all the combinations of signs. We will start with all '+' signs and then change one sign at a time.

First, all signs are plus:

$3 + 4 + 5 + 6$

Now, let us change the last sign to minus:

$3 + 4 + 5 - 6$

Next, we change the middle sign to minus, keeping the first sign plus:

$3 + 4 - 5 + 6$

$3 + 4 - 5 - 6$

Now, let us make the first sign minus:

$3 - 4 + 5 + 6$

$3 - 4 + 5 - 6$

$3 - 4 - 5 + 6$

$3 - 4 - 5 - 6$

We have listed all 8 possible combinations.

Answer

There are 8 different possibilities that exist, which are as follows: (i) $3 + 4 + 5 + 6$ (ii) $3 + 4 + 5 - 6$ (iii) $3 + 4 - 5 + 6$ (iv) $3 + 4 - 5 - 6$ (v) $3 - 4 + 5 + 6$ (vi) $3 - 4 + 5 - 6$ (vii) $3 - 4 - 5 + 6$ (viii) $3 - 4 - 5 - 6$