41) (a) A Venn diagram
b) P(RuF) = P(R)+P(F)-P(R-F)
P(only one sport)
e) P(Field Hockey/Not Rugby)=15/21
35) (a) P(A’) and P(B’)
(b) P(A) and P(B)
(c) At least one of the test detects steroid = 1-None shows presence of steroid
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)
7*5 ways – 5P48*7.
To eliminate zero, we fix zero in the first and rearrange the remaining number in the last four spaces. Hence, 4P4*7 such cases,
The total codes = 5P4*8*7)-4P4*7
The 2nd,3rd, 4th and 5th can be arranged in 4! Ways. Hence, number of arrangement = 4!x4!
1st – 4!
5th – 4!
2nd, 4th and 5th can be arranged into 3! ways.
Hence, The total codes=4!x4!x3!
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 3rd digit. The first digit can have either of 7 choices, excluding 5, 8 and 0. The 2nd 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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