7 choose 2

{ "query": { "display": "7C2", "symbolab_question": "#7C2" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "nCr", "subTopic": "Other", "default": "21" }, "steps": { "type": "interim", "title": "$$7\\:nCr\\:2:{\\quad}21$$", "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 $$n=7,\\:r=2$$", "result": "=\\frac{7!}{2!\\left(7-2\\right)!}" }, { "type": "interim", "title": "$$\\frac{7!}{2!\\left(7-2\\right)!}=21$$", "input": "\\frac{7!}{2!\\left(7-2\\right)!}", "result": "=21", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-2=5$$", "result": "=\\frac{7!}{2!\\cdot\\:5!}" }, { "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": [ "$$\\frac{7!}{5!}=7\\cdot\\:6$$" ], "result": "=\\frac{7\\cdot\\:6}{2!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{42}{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{42}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{42}{2}=21$$", "result": "=21" } ], "meta": { "solvingClass": "Solver" } } ] } }