By David Machin (auth.)

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4 . 1 . t h e b"~nom~a b 3 and b 5 ~n . 1v~ng . d t h e terms ~nvo F~n . expans~on of (a+ b) 5 • b 3 we set r . 1v~ng . 10, we obtain 5-3b3 ( 5) 3 a = 2 3 5! (5- 3)! 10 to obtain ( 55 )a5-5b5 a0b5 5! (5- 5)! 5! 2! = 3. 0! a We see in this latter calculation the need for the definition of O! 1. 5 In litters of mice from a certain cross, albinos occur in one in every three young. Calculate the probabilities of 0, 1, 2, 3, 4 and 5 albinos in litters of size 5 from such crosses. Let the proportion of albinos be Q = 1/3.

An are the coefficients of the polynomial and need to be specified if the graph is to be drawn. If n = 1, 2, 3, 4, then we have respectively linear, quadratic, cubic and quartic functions of x. 1 Show that the minimum value of the function y occurs when x = 3. Now y x 2 - 6x + 9 (x - 3) 2 hence if x = 3, y = 0. Any other value we give to x has an associated value of y > 0 and so, when x = 0, y is a minimum. 7) L and p the power equal to 2. 3 Graph of y (1 + x)- 1 p can take negative values. 7 becomes -1.

We should also note, again provided x is small, that the approximation may be used for non-integer and negative indices. 7. 995037. So the approximations are quite satisfactory. •• the exponential constant. Let us re-examine this limit but now by making use of the binomial expansion. 11 (a + b) n = an + na n-1 b + n(n -! 1) a n-2 b 2 2 + n(n- l)(n- 2) n-3b3 + ••• 3! 13) The number of terms on the right-hand side of this expression will increase with the index n and as n~ the number of terms in the binomial expansion will also tend to infinity.

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