1.04b Binomial probabilities: link to binomial expansion

6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$

11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant

1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
2. find the exact value of \(a\).

3

1. Find and simplify the first four terms in the expansion of \(( 2 - x ) ^ { 7 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(w ^ { 6 }\) in the expansion of \(\left( 2 - \frac { 1 } { 4 } w ^ { 2 } \right) ^ { 7 }\).

1

1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).

4

1. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
2. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).

1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.

1. 1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
   2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.

10

1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
   Give your answer in ascending powers of \(x\).
2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
   The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.

6

1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).