Section B: Probability and Statistics [41 marks] A code consists of 10 digits which are either zeros or ones, for example, 1011011010.

1. Apply the multiplication principle to find the total number of unrestricted codes

There are $10$ positions for the digits, and each position has $2$ independent choices (either $0$ or $1$).
$\text{Total number of codes} = 2^{10} = 1024$

1. Calculate the number of arrangements of 10 digits containing 4 identical zeros and 6 identical ones

$\text{Number of codes} = \frac{10!}{4!\,6!} \quad \text{(or } ^{10}C_4 \text{)}$
$= 210$

1. Set up the 'slotting' method to ensure all zeros are separated.

To ensure the $4$ zeros are separated, we first arrange the $6$ ones in a row. Since the ones are identical, there is only $1$ way to do this.

$\_ \; 1 \; \_ \; 1 \; \_ \; 1 \; \_ \; 1 \; \_ \; 1 \; \_ \; 1 \; \_$

This creates $7$ spaces in total ($5$ internal spaces between the ones, and $2$ spaces at the extreme ends) where the zeros can be placed.

2. Translate the condition 'first and last digits must be different'.

Since the digits are only zeros and ones, for the first and last digits to be different, the code must either start with $0$ and end with $1$, or start with $1$ and end with $0$.
In terms of the spaces created, this means exactly one of the $2$ end spaces must be chosen for a zero.

Number of ways to choose one end space = $^2C_1 = 2$

3. Place the remaining zeros and calculate the total combinations.

Once one end space is chosen, the remaining $3$ zeros must be placed in the $5$ internal spaces.

Number of ways to choose the internal spaces = $^5C_3 = 10$

Total number of such codes = $2 \times 10 = 20$

1. Identify the total number of unrestricted codes (from part ii).

\text{Total unrestricted arrangements} = \frac{10!}{4!6!} = 210

2. Calculate the number of codes where all 6 ones are together.

\text{Treat the } 6 \text{ ones as a single block. We are arranging this } 1 \text{ block and the } 4 \text{ identical zeros.}
\text{Number of arrangements} = \frac{(1 + 4)!}{4!} = \frac{5!}{4!} = 5

3. Calculate the number of codes where exactly 5 ones are together.

\text{Group } 5 \text{ ones into a block, leaving one separate } '1'.
\text{To ensure they do not combine to form } 6 \text{ ones together, they must be separated by at least one } '0'.
\text{The } 4 \text{ zeros create } 5 \text{ spaces between and around them: } _ 0 _ 0 _ 0 _ 0 _
\text{Place the block of } 5 \text{ ones and the separate } '1' \text{ into } 2 \text{ different spaces.}
\text{Number of arrangements} = 5 \times 4 = 20

4. Subtract the unwanted cases from the total to find the final number of valid codes.

\text{Number of codes with no more than } 4 \text{ ones together} = 210 - 5 - 20 = 185