Question Video: Finding the Sum of Binomial Coefficients | Nagwa Question Video: Finding the Sum of Binomial Coefficients | Nagwa

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

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In the expansion of (2π₯ + π)βΆ according to the descending powers of π₯, if the coefficient of the second term equals β1,344, find the value of π.

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### Video Transcript

In the expansion of two π₯ plus π all raised to the sixth power according to the descending powers of π₯, if the coefficient of the second term equals negative 1,344, find the value of π.

In this question, we are given a binomial expression in the form π plus π all raised to the πth power, where the power or exponent π is equal to six. We can find the general expansion of this by recalling Pascalβs triangle and the binomial theorem. Pascalβs triangle is a triangular array of the binomial coefficients. As the value of π in this case is six, our expansion will have seven terms. The first term is equal to one multiplied by π to the sixth power. The second term is equal to six multiplied by π to the fifth power multiplied by π or π to the first power. The third term is equal to 15 multiplied by π to the fourth power multiplied by π squared. This pattern continues as shown, where the power or exponent of π decreases and the exponent or power of π increases.

In this question, we need to expand two π₯ plus π to the sixth power. Therefore, the value of π from the general formula is two π₯ and the value of π is π. We are interested in the second term. This will be equal to six multiplied by two π₯ all to the fifth power multiplied by π to the first power or just π. As two to the fifth power is 32, two π₯ raised to the fifth power is 32π₯ to the fifth power. Multiplying this by six and by π gives us 192ππ₯ to the fifth power. The coefficient here is equal to 192π, and we are told in the question this is equal to negative 1,344. Clearing some space, we have 192π is equal to negative 1,344. Dividing both sides by 192 gives us a value of π equal to negative seven.

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