Question Video: Finding Binomial Coefficients | Nagwa Question Video: Finding Binomial Coefficients | Nagwa

Question Video: Finding Binomial Coefficients Mathematics • Third Year of Secondary School

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In the expansion of (2 − 𝑥²)⁹ according to the descending powers of 𝑥, find the coefficient of 𝑥¹⁰.

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Video Transcript

In the expansion of two minus 𝑥 squared all to the ninth power according to the descending powers of 𝑥, find the coefficient of 𝑥 to the 10th power.

In this question, we’re given a binomial expression written in the form 𝑎 plus 𝑏 all raised to the 𝑛th power. One way of solving a problem of this type is using Pascal’s triangle. However, when the exponent or power is large, usually greater than five, this process becomes quite cumbersome. As a result, we will 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. By considering the general term highlighted, we recall that 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. As we move from left to right, the powers of 𝑎 decrease and the powers of 𝑏 increase term by term.

In this question, the value of 𝑎 is two, the value of 𝑏 is negative 𝑥 squared, and 𝑛 is equal to nine. Whilst we could write out the full expansion of two minus 𝑥 squared all raised to the ninth power, we are only interested in the term containing 𝑥 to the 10th power. From our laws of exponents or indices, we recall that 𝑥 to the power of 𝑎 all raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. This means that 𝑥 squared raised to the fifth power is equal to 𝑥 to the 10th power. This means that the term we are dealing with contains 𝑏 to the fifth power. The term we are looking at in this question is nine choose five multiplied by 𝑎 to the fourth power multiplied by 𝑏 to the fifth power. Substituting in our values for 𝑎 and 𝑏, we have nine choose five multiplied by two to the fourth power multiplied by negative 𝑥 squared to the fifth power.

Nine choose five is equal to nine factorial divided by five factorial multiplied by four factorial. Typing this into the calculator gives us 126. Two to the fourth power is equal to 16. As negative one to the fifth power is negative one, negative 𝑥 squared to the fifth power is negative 𝑥 to the 10th power. Our expression simplifies to 126 multiplied by 16 multiplied by negative 𝑥 to the 10th power. This is equal to negative 2016 multiplied by 𝑥 to the 10th power. As we were asked to find the coefficient of 𝑥 to the 10th power, the final answer is negative 2016.

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