10 choose 6

{ "query": { "display": "10C6", "symbolab_question": "#10C6" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "nCr", "subTopic": "Other", "default": "210" }, "steps": { "type": "interim", "title": "$$10\\:nCr\\:6:{\\quad}210$$", "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 $$n=10,\\:r=6$$", "result": "=\\frac{10!}{6!\\left(10-6\\right)!}" }, { "type": "interim", "title": "$$\\frac{10!}{6!\\left(10-6\\right)!}=210$$", "input": "\\frac{10!}{6!\\left(10-6\\right)!}", "result": "=210", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$10-6=4$$", "result": "=\\frac{10!}{6!\\cdot\\:4!}" }, { "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": [ "$$\\frac{10!}{6!}=10\\cdot\\:9\\cdot\\:8\\cdot\\:7$$" ], "result": "=\\frac{10\\cdot\\:9\\cdot\\:8\\cdot\\:7}{4!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{5040}{4!}" }, { "type": "interim", "title": "$$4!=24$$", "input": "4!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$4!=1\\cdot\\:2\\cdot\\:3\\cdot\\:4$$" ], "result": "=1\\cdot\\:2\\cdot\\:3\\cdot\\:4" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2\\cdot\\:3\\cdot\\:4=24$$", "result": "=24" } ], "meta": { "solvingClass": "Solver" } }, { "type": "step", "result": "=\\frac{5040}{24}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{5040}{24}=210$$", "result": "=210" } ], "meta": { "solvingClass": "Solver" } } ] } }