c(13,4)

{ "query": { "display": "c(13, 4)", "symbolab_question": "#c(13,4)" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "nCr", "subTopic": "Other", "default": "715" }, "steps": { "type": "interim", "title": "$$13\\:nCr\\:4:{\\quad}715$$", "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 $$n=13,\\:r=4$$", "result": "=\\frac{13!}{4!\\left(13-4\\right)!}" }, { "type": "interim", "title": "$$\\frac{13!}{4!\\left(13-4\\right)!}=715$$", "input": "\\frac{13!}{4!\\left(13-4\\right)!}", "result": "=715", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$13-4=9$$", "result": "=\\frac{13!}{4!\\cdot\\:9!}" }, { "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": [ "$$\\frac{13!}{9!}=13\\cdot\\:12\\cdot\\:11\\cdot\\:10$$" ], "result": "=\\frac{13\\cdot\\:12\\cdot\\:11\\cdot\\:10}{4!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{17160}{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{17160}{24}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{17160}{24}=715$$", "result": "=715" } ], "meta": { "solvingClass": "Solver" } } ] } }