At least 8 of the words on the test are words that the student knows means that student knows 8 or 9 or 10 words.
1. Student knows 8 words (2 words are unknown): he knows the meanings of 14 words from a list of 22 words (8 words are unknown), then the number of ways to select 8 known words from 14 and 2 unknown words from 8 is
[tex] C_{14}^8\cdot C_{8}^2=\dfrac{14!}{8!(14-8)!} \cdot \dfrac{8!}{2!(8-2)!}=\dfrac{14!}{8!6!} \cdot \dfrac{8!}{2!6!} =\dfrac{8!\cdot 9\cdot 10 \cdot 11\cdot 12\cdot 13\cdot 14 }{8!\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6}\cdot \dfrac{6!\cdot 7\cdot 8}{6!\cdot 1\cdot2} =3\cdot 11\cdot 13\cdot 7\cdot 7\cdot 4=84084 [/tex].
2. Student knows 9 words (1 unknown): he knows the meanings of 14 words from a list of 22 words, then the number of ways to select 9 known words from 14 and 1 from 8 is
[tex] C_{14}^9\cdot C_8^1=\dfrac{14!}{9!(14-9)!}\cdot \dfrac{8!}{1!(8-1)!} =\dfrac{14!}{9!5!}\cdot \dfrac{8!}{1!7!} =\dfrac{9!\cdot 10 \cdot 11\cdot 12\cdot 13\cdot 14 }{9!\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5} \cdot \dfrac{7!\cdot 8}{7!}=11\cdot 13\cdot 14\cdot 8=16016 [/tex].
3. Student knows 10 words: he knows the meanings of 14 words from a list of 22 words, then the number of ways to select 10 known words from 14 is
[tex] C_{14}^{10}=\dfrac{14!}{10!(14-10)!} =\dfrac{14!}{10!4!} =\dfrac{10!\cdot 11\cdot 12\cdot 13\cdot 14 }{10!\cdot 1\cdot 2\cdot 3\cdot 4} =11\cdot 13\cdot 7=1001 [/tex].
4. He has [tex] C_{22}^{10}=\dfrac{22!}{10!(22-10)!} =\dfrac{22!}{12!10!}=\dfrac{12!\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18\cdot 19\cdot 20\cdot 21\cdot 22}{12!\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10} =13\cdot 7\cdot 17\cdot 19\cdot 22= 646646 [/tex] ways to select arbitrary 10 words from 22.
5.. The probability that at least 8 of the words on the test are words that the student knows is
[tex] Pr(\text{at least 8 words})=Pr(\text{8 words})+Pr(\text{9 words})+Pr(\text{10 words})=\dfrac{840846+16016+1001}{646646}=0.15634\approx 0.1569 [/tex]
