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334 ALGEBRA. 2. If four coins are tossed, find the chance that there should be 2 heads and 2 tails. 3. One of two events must happen : given that the chance of the one is two-thirds that of the other, find the odds in favor of the other. 4. Thirteen persons take their places at a round table. Show that it is 5 to 1 against 2 particular persons sitting together. 5. There are three events A, B, C, one of which must, and only one can, happen; the odds are 8 to 8 against A, 5 to 2 against B. Find the odds against C. 6. A has 3 shares in a lottery containing 3 prizes and 9 blanks; B has 2 shares in a lottery containing 2 prizes and 6 blanks. Compare their chances of success. 7. There are three works, one consisting of 3 volumes, one of 4, and the other of 1 volume. They are placed on a shelf at random. Prove that the chance that volumes of the same works are all together is 3 140. 8. The letters forming the word Clifton are placed at random in a row. What is the chance that the two vowels come together ? 9. In a hand at whist what is the chance that the four kings are held by a specified player. 10. There are 4 dollars and 3 half-dollars placed at random in a line. Show that the chance of the extreme coins being both half-dollars is 1 7.

MISCELLANEOUS EXAMPLES VI.

1. Simplify b-c a^2-(b-c)^2 + c-a b^2-(c -a)^2 + a-b c^2-(a-b)^2. 2. Extract the square root of (i.) 4x^4 + 6x^3 + 89 4 x^2 + 15x + 25. (ii.) x^8 - 2x^11 a^3 + 2a^4x^4 + x^14 a^6 - 2ax^7 + a^8. 3. A number of 3 digits exceeds 25 times the sum of the digits by 9; the middle digit increased by 3 is equal to the sum of the other digits, and the unit digit increased by 6 is equal to twice the sum of the other 2 digits: find the number.

4. Find the value of 2 3 2 + 3 2 3 -(2 3 3 1 5 ÷ 3 25 4)(1 2 6 - 24).

5. Solve (i.) 2 = 2+x + 2-x 2+x - 2-x (ii. ) 3x-11 - 3x= 12x - 23.