41) (a) A Venn diagram
b) P(RuF) = P(R)+P(F)-P(R-F)
=29/50+23/50-8/50
=44/50
=0.88
c) 14+18+6=38
P(only one sport)
=38/50
= 0.76
d) P(Volleyball/Rugby)
=P(VuR)/P(R)
=9/29
=0.31
e) P(Field Hockey/Not Rugby)=15/21
=5/7
=0.714
35) (a) P(A’) and P(B’)
=7/100x13/100
=91/10000
(b) P(A) and P(B)
=93/100x87/100
=8091/10000
(c) At least one of the test detects steroid = 1-None shows presence of steroid
= 1-91/10000
=9909/10000
6) a) We have to fill 4 spaces. We are left with 8 numbers to choose from (1,2,3,4,5,6,8,9)
4P_{4}*7
7*5 ways – 5P_{4}8*7.
To eliminate zero, we fix zero in the first and rearrange the remaining number in the last four spaces. Hence, 4P_{4}*7 such cases,
The total codes = 5P_{4}*8*7)-4P_{4}*7
=280-28
=252 codes
=5x4x3x2x1/(5-2)!
=120/6
=20 Codes
The 2^{nd},3^{rd}, 4^{th} and 5^{th} can be arranged in 4! Ways. Hence, number of arrangement = 4!x4!
=576codes.
1^{st} – 4!
5^{th} – 4!
2^{nd}, 4^{th} and 5^{th} can be arranged into 3! ways.
Hence, The total codes=4!x4!x3!
=576x6
23) Let 5 be the 1st digit. The other four digits can have either of the other 9 choices, as we have excluded 8. This gives 9^4 numbers.
Let 5 be the 2nd digit. The first digit can have either of 7 choices, excluding 5, 8 and 0. The other three digits can have any of the 9 choices. This gives 7 * 9^3 numbers.
Let 5 be the 3^{rd} digit. The first digit can have either of 7 choices, excluding 5, 8 and 0. The 2^{nd} digit can have 8 choices excluding 5 and 8. The other two digits can have any of the 9 choices. This gives 7 * 8 *9^2 numbers.
Let 5 be the 4th digit. The first digit can have either of 7 choices, excluding 5, 8 and 0. The 2nd and 3rd digits can have any of 8 choices. This gives 7 * 8^2*9 numbers.
Let 5 be the 5th digit. The first digit can have either of 7 choices, excluding 5, 8 and 0. The 2nd, 3rd and 4th digits can have any of 8 choices. This gives 7 * 8^3 numbers.
Hence, the total possible numbers;
= 9^4 + 7*9^3 + 7*8*9^2 + 7*8^2*9 + 7*8^3= 23816
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