Question Video: Finding Binomial Coefficients Mathematics

Video Transcript

Find the coefficient of 𝑥 cubed in the expansion of two plus three 𝑥 all to the eighth power.

This is an example of a binomial expression written in the form 𝑎 plus 𝑏 all raised to the 𝑛th power. We can solve problems of this type using Pascal’s triangle. However, as the power or exponent in this case is larger than five, it would be very time-consuming to write out every row of Pascal’s triangle. We will therefore use the binomial theorem. This states that 𝑎 plus 𝑏 to the 𝑛th power is equal to 𝑛 choose zero multiplied by 𝑎 to the 𝑛th power plus 𝑛 choose one multiplied by 𝑎 to the 𝑛 minus oneth power multiplied by 𝑏 to the first power, and so on, where 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. As we move along term by term, the powers or exponents of 𝑛 decrease, whereas the powers or exponents of 𝑏 increase.

In this question, the value of 𝑎 is two, 𝑏 is equal to three 𝑥, and the power or exponent 𝑛 equals eight. At this stage, we could write out the whole expansion. However, we are only interested in the term containing 𝑥 cubed. As 𝑏 is equal to three 𝑥, this will be the term in the general expansion containing 𝑏 cubed. This term is equal to eight choose three multiplied by 𝑎 to the fifth power multiplied by 𝑏 cubed.

Substituting in our values of 𝑎 and 𝑏, we have eight choose three multiplied by two to the fifth power multiplied by three 𝑥 all cubed. Eight choose three is equal to eight factorial divided by five factorial multiplied by three factorial. We recall that eight factorial can be rewritten as eight multiplied by seven multiplied by six multiplied by five factorial. We can then cancel five factorial from the numerator and denominator. Three factorial is equal to six. We can therefore cancel this from the numerator and denominator. Eight choose three is therefore equal to eight multiplied by seven, which is equal to 56.

Alternatively, we could’ve just typed this straight in to our calculator. Two to the fifth power is equal to 32. As three cubed is equal to 27, three 𝑥 all cubed is 27𝑥 cubed. Our expression becomes 56 multiplied by 32 multiplied by 27𝑥 cubed. This is equal to 48,384𝑥 cubed. As we just want the coefficient of 𝑥 cubed, the final answer is 48,384.