12 choose 10

{ "query": { "display": "12C10", "symbolab_question": "#12C10" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "nCr", "subTopic": "Other", "default": "66" }, "steps": { "type": "interim", "title": "$$12\\:nCr\\:10:{\\quad}66$$", "steps": [ { "type": "definition", "title": "n choose r", "text": "Gives the number of subsets of r elements, out of n elements<br/>$$nCr=\\frac{n!}{r!\\left(n-r\\right)!}$$" }, { "type": "step", "result": "=\\frac{n!}{r!\\left(n-r\\right)!}" }, { "type": "step", "primary": "Plug in $$n=12,\\:r=10$$", "result": "=\\frac{12!}{10!\\left(12-10\\right)!}" }, { "type": "interim", "title": "$$\\frac{12!}{10!\\left(12-10\\right)!}=66$$", "input": "\\frac{12!}{10!\\left(12-10\\right)!}", "result": "=66", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$12-10=2$$", "result": "=\\frac{12!}{2!\\cdot\\:10!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{12!}{10!}=12\\cdot\\:11$$" ], "result": "=\\frac{12\\cdot\\:11}{2!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{132}{2!}" }, { "type": "interim", "title": "$$2!=2$$", "input": "2!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$2!=1\\cdot\\:2$$" ], "result": "=1\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver" } }, { "type": "step", "result": "=\\frac{132}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{132}{2}=66$$", "result": "=66" } ], "meta": { "solvingClass": "Solver" } } ] } }