By G. R. Veena
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Additional resources for Comprehensive Basic Mathematics: v. II
Number of selections = ( a) 6 c4 × 9 c6 = 1260 ( b) 6 c5 × 9 c5 = 756 (c ) 6 c6 × 9 c 4 = 126 So Total number of selections = (iv) As ( a) ( b) (c ) ( d) (By fundamental principle) 1260 + 756 + 126 = 2142. there is to be atmost 7 teachers, the committee may consist of 7 teachers and 3 students 6 teachers and 4 students 5 teachers and 5 students 4 teachers and 6 students. Permutation and Combination 41 ∴ Number of selections = (a ) 9 c7 × 6 c3 = 720 (b ) 9 c6 × 6 c 4 = 1260 (c ) 9 c5 × 6 c5 = 756 (d ) 9 c4 × 6 c6 = 126 ∴ Total number of selections = 720 + 1260 + 756 + 126 = 2862.
Number of numbers = 6 6×5× 4 = 2⋅ 3 2 = 60. Now keeping 2 in the million’s place and arranging the rest. There are 6 numbers 1, 2, 3, 3, 3, 4 in which 3 occurs thrice. ∴ Number of numbers = 6 =6×5×4 3 = 120. Total number of numbers less than 3 millions. = 60 + 120 = 180. 7 CIRCULAR PERMUTATION: If we have to arrange 4 letters A, B, C, D in a row. Then 2 of the arrangements would be ABCD, BCDA and we treat these two arrangements as different. But if we arrange along the circumference of the circle, the two arrangements and are one and the same.
5. VI. 1. 2. 3. 4. 5. 6. 7. 8. Tautology Tautology Contradiction p∨ ~ p ∧ q T T T f 7. p→q∧r T T T T T F T T T 2. Tautology 6. Contradiction 10. Tautology No, not logically equivalent. Logically equivalent. Logically equivalent. f F a p ∧ qf ↔ a q∧ ~ r f F T T T T F ~ p →~ q T F T 3. Contradiction 7. Neither F F 4. Neither 8. Neither 2. Logically equivalent. 4. No, not logically equivalent. 6. Logically equivalent. It is not odd or 6 is not even Cow is big and it is not black. 2 lines are parallel and they intersect I will pass the examination and the questions are not easy or I will not pass the examination and the questions are easy.