<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>2419401 - Disclosure - CONTINGENCIES (Details)</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption>In Millions, unless otherwise specified</RoundingOption><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode /><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol /><contextRef><ContextID>D2013Q2YTD</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0001452575</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-01-01T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-30T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS><UnitProperty><UnitID>Proceeding</UnitID><UnitType>Standard</UnitType><StandardMeasure><MeasureSchema>http://www.meadjohnson.com/20130630</MeasureSchema><MeasureValue>Proceeding</MeasureValue><MeasureNamespace>mjn</MeasureNamespace><MeasureDescription /><MeasureType /></StandardMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode /><OriginalCurrencyCode /></MCU><CurrencySymbol /><Labels><Label Key="CalendarSupplement" Id="0" Label="6 Months Ended" /><Label Key="Calendar" Id="1" Label="Jun. 30, 2013" /><Label Key="" Id="2" Label="Proceeding" /></Labels></Column><Column FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode>USD</CurrencyCode><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol>$</CurrencySymbol><contextRef><ContextID>D2012Q4YTD</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0001452575</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2012-01-01T00:00:00</PeriodStartDate><PeriodEndDate>2012-12-31T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS><UnitProperty><UnitID>usd</UnitID><UnitType>Standard</UnitType><StandardMeasure><MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema><MeasureValue>USD</MeasureValue><MeasureNamespace>iso4217</MeasureNamespace><MeasureDescription /><MeasureType /></StandardMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode>USD</CurrencyCode><OriginalCurrencyCode>USD</OriginalCurrencyCode></MCU><CurrencySymbol>$</CurrencySymbol><Labels><Label Key="CalendarSupplement" Id="0" Label="12 Months Ended" /><Label Key="Calendar" Id="1" Label="Dec. 31, 2012" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_SubsequentEventLineItems</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="" UnitID=""><Id>2</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Subsequent Event [Line Items]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>4</Level><ElementName>mjn_ChinaAntitrustReview</ElementName><ElementPrefix>mjn_</ElementPrefix><IsBaseElement>false</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2013Q2YTD" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText>As previously announced, the Company is fully cooperating with the Bureau of Price Supervision and Anti-Monopoly of China's National Development and Reform Commission (&#x201C;NDRC&#x201D;) in its recent antitrust review of the infant formula industry's practices relating to resale prices (&#x201C;China Antitrust Review&#x201D;).&#xA0; Accordingly, the Company agreed to reduce the ex-factory price of certain products to its direct customers in China. The Company cannot predict what the NDRC's final resolution of this matter will be.  Under China law, the NDRC has complete discretion, among other actions, to 1) issue an order prohibiting future conduct deemed illegal; and/or 2) confiscate proceeds deemed illegally obtained; and/or 3) assess a penalty in the range of 1 - 10% of a company's prior year sales in China. In 2012, MJN recorded approximately $800 million of sales in China.
</NonNumbericText><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="" UnitID=""><Id>2</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><ElementDefenition>China Antitrust Review

As previously announced, the Company is fully cooperating with the Bureau of Price Supervision and Anti-Monopoly of China's National Development and Reform Commission (&#x201C;NDRC&#x201D;) in its recent antitrust review of the infant formula industry's practices relating to resale prices (&#x201C;China Antitrust Review&#x201D;).  Accordingly, the Company agreed to reduce the ex-factory price of certain products to its direct customers in China. The Company cannot predict what the NDRC's final resolution of this matter will be.  Under China law, the NDRC has complete discretion, among other actions, to 1) issue an order prohibiting future conduct deemed illegal; and/or 2) confiscate proceeds deemed illegally obtained; and/or 3) assess a penalty in the range of 1 - 10% of a company's prior year sales in China. In 2012, MJN recorded approximately $800 million of sales in China.



In accordance with Accounting Standards Codification 450-20, the Company has recorded a charge related to the China Antitrust Review. This charge did not have a material effect on the financial statements for the quarter ended June 30, 2013. However, the NDRC's final resolution of the matter could result in an incremental charge in excess of the amount recorded at June 30, 2013 and could have a material impact on the Company's operating results in the quarter in which such resolution is obtained. Based on the range of possible remedies under China law as described above, the Company determined that it cannot estimate a reasonably possible range for such a potential incremental charge. However, the Company believes that there would not be a material effect on its business, financial condition or cash flows if such an incremental charge were to be incurred.</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>China Antitrust Review</Label></Row><Row FlagID="0"><Id>3</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>4</Level><ElementName>mjn_Approx2012SalesinChina</ElementName><ElementPrefix>mjn_</ElementPrefix><IsBaseElement>false</IsBaseElement><BalanceType>credit</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="D2012Q4YTD" UnitID="usd"><Id>2</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>800000000</NumericAmount><RoundedNumericAmount>800</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>Approximate 2012 Sales in China</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Approx2012SalesinChina</Label></Row><Row FlagID="0"><Id>4</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>mjn_LossContingencyNumberOfProceedingsThatCouldHaveMaterialAdverseAffectMinimum</ElementName><ElementPrefix>mjn_</ElementPrefix><IsBaseElement>false</IsBaseElement><BalanceType>na</BalanceType><PeriodType>instant</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="I2013Q2" UnitID="Proceeding"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>1</NumericAmount><RoundedNumericAmount>1</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell><Cell FlagID="0" ContextID="" UnitID=""><Id>2</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:integerItemType</ElementDataType><SimpleDataType>integer</SimpleDataType><ElementDefenition>The minimum number of proceedings that could have a material adverse affect on the Company's results of operations for the periods in which a loss is recognized.</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>256</UnitID><Label>Minimum number of proceedings that could have a material impact</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>CONTINGENCIES (Details) (USD $)</ReportName><MonetaryRoundingLevel>Millions</MonetaryRoundingLevel><SharesRoundingLevel>UnKnown</SharesRoundingLevel><PerShareRoundingLevel>UnKnown</PerShareRoundingLevel><ExchangeRateRoundingLevel>UnKnown</ExchangeRateRoundingLevel><HasCustomUnits>true</HasCustomUnits><IsEmbedReport>false</IsEmbedReport><IsMultiCurrency>false</IsMultiCurrency><ReportType>Sheet</ReportType><RoleURI>http://www.meadjohnson.com/role/ContingenciesDetails</RoleURI><NumberOfCols>2</NumberOfCols><NumberOfRows>4</NumberOfRows></InstanceReport>
