The following financial information relates to Pink Limited in $2023$:

Profits before tax: $$426,000,000$

Income tax expense: $$(84,000,000)$

Profits after tax: $$342,000,000$

Less: Ordinary dividends: $$(96,000,000)$

Less: Preference dividends: $$(12,000,000)$

Retained profits for the year: $$234,000,000$

Retained profits at $1$ January $2023$: $$157,200,000$

Retained profits at $31$ December $2023$: $$391,200,000$

Number of issued ordinary shares at $1$ January $2023$: $240,000,000$shares

Number of issued preference shares at $1$ January $2023$: $6,000,000$shares

For the year ended $31$ December $2023,$ determine the basic earnings per share $($EPS$)$ and diluted EPS $($if appropriate$)$ for Pink Limited based on the following independent assumptions:

a$)$ Determine the basic and diluted EPS $($if appropriate$)$ assuming there was no change in the issued share capital in $2023$ for Pink Limited.

b$)$ Determine the basic and diluted EPS $($if appropriate$)$ assuming Pink Limited made a rights issue of ordinary shares in a proportion of one for every four shares currently held at $$3$ per share on $1$ August $2023.$ The market price of the ordinary share immediately before the rights issue was $$4$ per share.

c$)$ Determine the basic and diluted EPS $($if appropriate$)$ assuming there was no change in the issued share capital in $2023$ for Pink Limited. Additionally, on $1$ January $2023,500,000$ options with an exercise price of $$15$ were issued. No options were exercised in $2023.$ Furthermore, on $1$ January $2023,5\%$ convertible loan stocks with a nominal value of $$10,000,000$ were issued. Each $$1,000$ loan stock could be converted to $50$ ordinary shares. No convertible loan stocks were exercised in $2023.$ The tax rate for the year $2022$ was $20\%.$ The average share price of ordinary shares during $2022$ was $$20$ per share.