37","title":"1.6.1 Binomial expansion for positive integer n Flashcards","topicId":"8de5f1da-0b71-41f1-a57c-981a54219233","topicSlug":"binomial-expansion-for-positive-integer-n","cards":[{"id":"16c6fda5-a8a2-4792-85dd-fac18b33ba15","cardType":"basic","front":"What does $0!$ equal?","back":"$1$","points":1,"mcqOptions":null},{"id":"06a54071-2a22-4474-b061-7301a0918703","cardType":"basic","front":"Calculate $\\binom{7}{3}$.","back":"$35$","points":1,"mcqOptions":null},{"id":"3d8e345c-8aa7-4568-934d-05836bd964de","cardType":"basic","front":"How many terms are in the complete expansion of $(a+b)^n$?","back":"$n+1$ terms","points":1,"mcqOptions":null},{"id":"e3a0e610-c2f4-4299-bcdc-cc09396b6249","cardType":"cloze","front":"The symmetry property of binomial coefficients is $\\text{[...]}$.","back":"The symmetry property of binomial coefficients is $\\boxed{\\binom{n}{r}=\\binom{n}{n-r}}$.","points":null,"mcqOptions":null},{"id":"c4ad57c0-e1f0-4b63-82db-244738fa60c7","cardType":"basic","front":"How do the powers change between consecutive terms of $(a+b)^n$?","back":"The power of $a$ decreases by 1; the power of $b$ increases by 1.","points":2,"mcqOptions":null},{"id":"d4c3c84c-ca82-4a86-80f3-64627b3bded0","cardType":"basic","front":"What must the powers of $a$ and $b$ add to in each term of $(a+b)^n$?","back":"They add to $n$.","points":1,"mcqOptions":null},{"id":"690a01b0-6b6c-4637-bf8a-c02c1d59669e","cardType":"cloze","front":"For positive integer $n$, $(a+b)^n=$ $\\text{[...]}$.","back":"For positive integer $n$, $(a+b)^n=$ $\\boxed{\\displaystyle\\sum_{r=0}^{n}\\binom{n}{r}a^{n-r}b^r}$.","points":null,"mcqOptions":null},{"id":"2458688a-2198-4070-81fb-1afcd0c06db5","cardType":"basic","front":"Why is $(2x)^3$ equal to $8x^3$, not $2x^3$?","back":"The power applies to both factors: $2^3x^3=8x^3$.","points":2,"mcqOptions":null},{"id":"3fd5f733-39e7-4635-8998-a796039a84f0","cardType":"basic","front":"Predict the sign of $(-bx)^r$ when $r$ is odd.","back":"Negative","points":1,"mcqOptions":null},{"id":"58bdb698-021b-4aef-926c-2a0c828b0967","cardType":"basic","front":"Predict the sign of $(-bx)^r$ when $r$ is even.","back":"Positive","points":1,"mcqOptions":null},{"id":"03de0b3d-5aaf-445a-9588-3b77c34f7154","cardType":"cloze","front":"The term containing $x^r$ in $(a+bx)^n$ is $\\text{[...]}$.","back":"The term containing $x^r$ in $(a+bx)^n$ is $\\boxed{T_{r+1}=\\binom{n}{r}a^{n-r}b^rx^r}$.","points":null,"mcqOptions":null},{"id":"1f8192cb-bdab-428f-a76e-5bbf7b369641","cardType":"basic","front":"Which term number contains $x^3$ in an expansion of $(a+bx)^n$?","back":"The fourth term.","points":1,"mcqOptions":null},{"id":"b8c1ac76-d4bc-4b11-8799-f03cb6293669","cardType":"cloze","front":"The coefficient of $x^r$ in $(a+bx)^n$ is $\\text{[...]}$.","back":"The coefficient of $x^r$ in $(a+bx)^n$ is $\\boxed{\\binom{n}{r}a^{n-r}b^r}$.","points":null,"mcqOptions":null},{"id":"76d074c5-4d51-46fa-97fd-a8d41601f1c0","cardType":"basic","front":"Expand $(2+x)^4$.","back":"$16+32x+24x^2+8x^3+x^4$","points":2,"mcqOptions":null},{"id":"fccbf199-f81a-4200-9be2-e8de02643926","cardType":"basic","front":"Expand $(1-3x)^4$.","back":"$1-12x+54x^2-108x^3+81x^4$","points":2,"mcqOptions":null},{"id":"a2e76b8b-bcf5-494f-be1b-db566704753e","cardType":"basic","front":"What constant term should appear in the expansion of $(a+bx)^n$?","back":"$a^n$","points":1,"mcqOptions":null},{"id":"7316b12b-358e-403d-bd15-795cd9dfde52","cardType":"basic","front":"What should the coefficient of $x^n$ be in $(a+bx)^n$?","back":"$b^n$","points":1,"mcqOptions":null},{"id":"0c5c2d7a-a9a9-4a9c-aaaa-4f2e381bc196","cardType":"cloze","front":"For $(a+bx)^n$, the sum of the coefficients is $\\text{[...]}$.","back":"For $(a+bx)^n$, the sum of the coefficients is $\\boxed{(a+b)^n}$.","points":null,"mcqOptions":null},{"id":"ddd9a7f0-34fc-41b2-aa2a-d8