A college offers 41 different subjects including 9 different languages - OCR - GCSE Maths - Question 14 - 2023 - Paper 4

Next, we need to calculate the total combinations that do not include any languages.

• For Option A (14 subjects, 2 languages): Non-language subjects = 14 - 2 = 12

• For Option B (12 subjects, 3 languages): Non-language subjects = 12 - 3 = 9

• For Option C (15 subjects, 4 languages): Non-language subjects = 15 - 4 = 11

Thus, the combinations without any language would be:

$$ext{Combinations without any language} = 12 imes 9 imes 11 = 1188$$

Now, we find the combinations that include at least one language by subtracting the combinations without any language from the total combinations:

$$ext{Combinations with at least one language} = ext{Total combinations} - ext{Combinations without any language} \ = 2520 - 1188 = 1332$$

Finally, we compute the proportion of combinations that include at least one language:

ext{Proportion} = rac{ ext{Combinations with at least one language}}{ ext{Total combinations}} \ = rac{1332}{2520} = rac{111}{210} ext{ (after simplification)}

Thus, the proportion of subject combinations that include at least one language is

rac{111}{210}.